Class Bsc Book Statistics All Chpter Wise Notes 50s1f

Statistics: Introduction chp1 Definitions Statistics Collection of methods for planning experiments, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions. Variable Characteristic or attribute that can assume different values Random Variable A variable whose values are determined by chance. Population All subjects possessing a common characteristic that is being studied. Sample A subgroup or subset of the population. Parameter Characteristic or measure obtained from a population. Statistic (not to be confused with Statistics) Characteristic or measure obtained from a sample. Descriptive Statistics Collection, organization, summarization, and presentation of data. Inferential Statistics Generalizing from samples to populations using probabilities. Performing hypothesis testing, determining relationships between variables, and making predictions. Qualitative Variables Variables which assume non-numerical values. Quantitative Variables Variables which assume numerical values. Discrete Variables Variables which assume a finite or countable number of possible values. Usually obtained by counting. Continuous Variables Variables which assume an infinite number of possible values. Usually obtained by measurement. Nominal Level

Level of measurement which classifies data into mutually exclusive, all inclusive categories in which no order or ranking can be imposed on the data. Ordinal Level Level of measurement which classifies data into categories that can be ranked. Differences between the ranks do not exist. Interval Level Level of measurement which classifies data that can be ranked and differences are meaningful. However, there is no meaningful zero, so ratios are meaningless. Ratio Level Level of measurement which classifies data that can be ranked, differences are meaningful, and there is a true zero. True ratios exist between the different units of measure. Random Sampling Sampling in which the data is collected using chance methods or random numbers. Systematic Sampling Sampling in which data is obtained by selecting every kth object. Convenience Sampling Sampling in which data is which is readily available is used. Stratified Sampling Sampling in which the population is divided into groups (called strata) according to some characteristic. Each of these strata is then sampled using one of the other sampling techniques. Cluster Sampling Sampling in which the population is divided into groups (usually geographically). Some of these groups are randomly selected, and then all of the elements in those groups are selected.

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Statistics: Introduction

Population vs Sample The population includes all objects of interest whereas the sample is only a portion of the population. Parameters are associated with populations and statistics with samples.

Parameters are usually denoted using Greek letters (mu, sigma) while statistics are usually denoted using Roman letters (x, s). There are several reasons why we don't work with populations. They are usually large, and it is often impossible to get data for every object we're studying. Sampling does not usually occur without cost, and the more items surveyed, the larger the cost. We compute statistics, and use them to estimate parameters. The computation is the first part of the statistics course (Descriptive Statistics) and the estimation is the second part (Inferential Statistics)

Discrete vs Continuous Discrete variables are usually obtained by counting. There are a finite or countable number of choices available with discrete data. You can't have 2.63 people in the room. Continuous variables are usually obtained by measuring. Length, weight, and time are all examples of continous variables. Since continuous variables are real numbers, we usually round them. This implies a boundary depending on the number of decimal places. For example: 64 is really anything 63.5 <= x < 64.5. Likewise, if there are two decimal places, then 64.03 is really anything 63.025 <= x < 63.035. Boundaries always have one more decimal place than the data and end in a 5.

Levels of Measurement There are four levels of measurement: Nominal, Ordinal, Interval, and Ratio. These go from lowest level to highest level. Data is classified according to the highest level which it fits. Each additional level adds something the previous level didn't have.    

Nominal is the lowest level. Only names are meaningful here. Ordinal adds an order to the names. Interval adds meaningful differences Ratio adds a zero so that ratios are meaningful.

Types of Sampling There are five types of sampling: Random, Systematic, Convenience, Cluster, and Stratified.

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Random sampling is analogous to putting everyone's name into a hat and drawing out several names. Each element in the population has an equal chance of occuring. While this is the preferred way of sampling, it is often difficult to do. It requires that a complete list of every element in the population be obtained. Computer generated lists are often used with random sampling. You can generate random numbers using the TI82 calculator. Systematic sampling is easier to do than random sampling. In systematic sampling, the list of elements is "counted off". That is, every kth element is taken. This is similar to lining everyone up and numbering off "1,2,3,4; 1,2,3,4; etc". When done numbering, all people numbered 4 would be used. Convenience sampling is very easy to do, but it's probably the worst technique to use. In convenience sampling, readily available data is used. That is, the first people the surveyor runs into. Cluster sampling is accomplished by dividing the population into groups -usually geographically. These groups are called clusters or blocks. The clusters are randomly selected, and each element in the selected clusters are used. Stratified sampling also divides the population into groups called strata. However, this time it is by some characteristic, not geographically. For instance, the population might be separated into males and females. A sample is taken from each of these strata using either random, systematic, or convenience sampling.

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TI-82: Generating Random Numbers You can generate random numbers on the TI-82 calculator using the following sequence. N is the number of different values which could be and S is the minimum number. int (N*rand+S)

INT is found under the MATH menu (math num 4). RAND is also found under the MATH menu (math prb 1). Simulate the rolling of a die (1-6): int

(6*rand+1)

Simulate the flipping of a coin (0-1): int

(2*rand)

This works because the rand function returns a random number between 0 and 1 (including 0 but not including 1). When it is multiplied by N, it becomes between 0 and N, and then S is added, so it becomes between S and S+N. If you have two values (A and B) that you need random numbers between, then you can generate them using the following formulas. N=B-A+1 int (N*rand+A)

Notice it is B-A+1 not B-A. Everyone agrees there are 10 numbers between 1 and 10 (inclusive). But, if you take 10-1, you get 9, not 10. Also, in the formula above, replace the N by the actual number of different values. Since the calculator re the last formula put in, and evaluates it when you hit enter, to generate more random numbers, just hit enter again. Each time you hit enter, you will get another random number.

Sampling Lab The purpose of this laboratory exercise is to familiarize yourself with the different sampling techniques. You need one page from a movie listing (like contained in TV-Guide). Note, if you actually use TV Guide®, then you need to use two facing pages. Pick a page with little extraneous material, other than the listings, on it. For the purposes of this sampling project, a movie is included on the page or in a cluster if the running time for the movie falls on the page. Random Sampling

Number each movie on the page. If there are a lot of movies, you may wish to number every other or every third movie. Generate a random sample on 8 numbers between 1 and the number of movies on the page. Write down the # generated and the running time for the movie corresponding to that number.

Systematic Sampling

Generate a random number between 1 and 6. Beginning with the movie corresponding to that number, and then taking every 6th movie thereafter, write the # of the movie and the running length of the movie. Convenience Sampling

Write down the running time of the first eight movies. Stratified Sampling

On a separate piece of paper, write down the running times of all PG/PG13, R, and not-rated (either NR or no rating given) movies in three columns -- ignore all other types (NC17, G, etc). Split a sample of 8 proportionally to each type of movie (if R is 40%, then sample 40% of 8 = 3.2 -> 3 R movies). Use random sampling within each movie type. Record the running lengths of the movies selected. Cluster Sampling

Divide the page into equal regions so that each region has roughly 3 - 4 movies in each cluster. Randomly select 3 clusters, and record the running length of all movies in those clusters.

Statistics: Frequency Distributions & Graphs chp 2

Definitions Raw Data Data collected in original form. Frequency The number of times a certain value or class of values occurs. Frequency Distribution The organization of raw data in table form with classes and frequencies. Categorical Frequency Distribution

A frequency distribution in which the data is only nominal or ordinal. Ungrouped Frequency Distribution A frequency distribution of numerical data. The raw data is not grouped. Grouped Frequency Distribution A frequency distribution where several numbers are grouped into one class. Class Limits Separate one class in a grouped frequency distribution from another. The limits could actually appear in the data and have gaps between the upper limit of one class and the lower limit of the next. Class Boundaries Separate one class in a grouped frequency distribution from another. The boundaries have one more decimal place than the raw data and therefore do not appear in the data. There is no gap between the upper boundary of one class and the lower boundary of the next class. The lower class boundary is found by subtracting 0.5 units from the lower class limit and the upper class boundary is found by adding 0.5 units to the upper class limit. Class Width The difference between the upper and lower boundaries of any class. The class width is also the difference between the lower limits of two consecutive classes or the upper limits of two consecutive classes. It is not the difference between the upper and lower limits of the same class. Class Mark (Midpoint) The number in the middle of the class. It is found by adding the upper and lower limits and dividing by two. It can also be found by adding the upper and lower boundaries and dividing by two. Cumulative Frequency The number of values less than the upper class boundary for the current class. This is a running total of the frequencies. Relative Frequency

The frequency divided by the total frequency. This gives the percent of values falling in that class. Cumulative Relative Frequency (Relative Cumulative Frequency) The running total of the relative frequencies or the cumulative frequency divided by the total frequency. Gives the percent of the values which are less than the upper class boundary. Histogram A graph which displays the data by using vertical bars of various heights to represent frequencies. The horizontal axis can be either the class boundaries, the class marks, or the class limits. Frequency Polygon A line graph. The frequency is placed along the vertical axis and the class midpoints are placed along the horizontal axis. These points are connected with lines. Ogive A frequency polygon of the cumulative frequency or the relative cumulative frequency. The vertical axis the cumulative frequency or relative cumulative frequency. The horizontal axis is the class boundaries. The graph always starts at zero at the lowest class boundary and will end up at the total frequency (for a cumulative frequency) or 1.00 (for a relative cumulative frequency). Pareto Chart A bar graph for qualitative data with the bars arranged according to frequency. Pie Chart Graphical depiction of data as slices of a pie. The frequency determines the size of the slice. The number of degrees in any slice is the relative frequency times 360 degrees. Pictograph A graph that uses pictures to represent data.

Stem and Leaf Plot A data plot which uses part of the data value as the stem and the rest of the data value (the leaf) to form groups or classes. This is very useful for sorting data quickly.

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Statistics: Grouped Frequency Distributions

Guidelines for classes 1. There should be between 5 and 20 classes. 2. The class width should be an odd number. This will guarantee that the class midpoints are integers instead of decimals. 3. The classes must be mutually exclusive. This means that no data value can fall into two different classes 4. The classes must be all inclusive or exhaustive. This means that all data values must be included. 5. The classes must be continuous. There are no gaps in a frequency distribution. Classes that have no values in them must be included (unless it's the first or last class which are dropped). 6. The classes must be equal in width. The exception here is the first or last class. It is possible to have an "below ..." or "... and above" class. This is often used with ages. Creating a Grouped Frequency Distribution

1. 2. 3. 4.

Find the largest and smallest values Compute the Range = Maximum - Minimum Select the number of classes desired. This is usually between 5 and 20. Find the class width by dividing the range by the number of classes and rounding up. There are two things to be careful of here. You must round up, not off. Normally 3.2 would round to be 3, but in rounding up, it becomes 4. If

the range divided by the number of classes gives an integer value (no remainder), then you can either add one to the number of classes or add one to the class width. Sometimes you're locked into a certain number of classes because of the instructions. The Bluman text fails to mention the case when there is no remainder. 5. Pick a suitable starting point less than or equal to the minimum value. You will be able to cover: "the class width times the number of classes" values. You need to cover one more value than the range. Follow this rule and you'll be okay: The starting point plus the number of classes times the class width must be greater than the maximum value. Your starting point is the lower limit of the first class. Continue to add the class width to this lower limit to get the rest of the lower limits. 6. To find the upper limit of the first class, subtract one from the lower limit of the second class. Then continue to add the class width to this upper limit to find the rest of the upper limits. 7. Find the boundaries by subtracting 0.5 units from the lower limits and adding 0.5 units from the upper limits. The boundaries are also half-way between the upper limit of one class and the lower limit of the next class. Depending on what you're trying to accomplish, it may not be necessary to find the boundaries. 8. Tally the data. 9. Find the frequencies. 10. Find the cumulative frequencies. Depending on what you're trying to accomplish, it may not be necessary to find the cumulative frequencies. 11. If necessary, find the relative frequencies and/or relative cumulative frequencies. It is possible to have the TI-82 calculator find the frequencies for you. You will have to find the class width and class boundaries first.

TI-82: Lists and Statistics There are two features of the TI-82 calculator that will be used. Lists and Statistics. The STAT key is located at the top center of the calculator and the LIST key is obtained by 2nd STAT. There are six lists that you can work with at any time on the calculator. Each set of data requires a list. If you include frequencies for the frequency distribution, then it will require a list for the data and a separate list for the

frequencies. The lists are labeled L1, L2, L3, L4, L5, and L6 and are accessed on the calculator by pressing "2nd 1", "2nd 2", etc.

STATS Key STAT has two major categories, EDIT and CALC STAT-EDIT

1. Edit - Use this to enter data into a list. 2. SortA( - This will sort a list in ascending order. This is useful if you want to find the frequencies after you have already established the limits or boundaries. Since the data is sorted in order, you just have to go through and count the number in each class. You don't need to do the tally. This will replace the list you tell it sort. 3. SortD( - This will sort a list in descending order. This will replace the list you tell it to sort. 4. ClrList - This will erase any existing lists. STAT-CALC

1. 1-Var Stats - This is used when there is only one variable. It will handle both raw data and frequency distributions. 2. 2-Var Stats - This is used when there are two variables, and x and y. This won't happen until the end of the semester. 3. Setup - You will need to check the setup before you find any other statistical values from this menu. It allows you to specify which list(s) you put the data into and if necessary, which list contains the frequencies. 4. Med-Med - A regression model that isn't used in this course. 5. LinReg(ax+b) - A regression model that will be used later in the course after we talk about two variable statistics. 6. QuadReg - A regression model that isn't used in this course. 7. CubicReg - A regression model that isn't used in this course. 8. QuartReg - A regression model that isn't used in this course. 9. LinReg(ax+b) - A regression model that will be used later in the course after we talk about two variable statistics. The book uses this model, however, we will use #5 instead. 10. LnReg - A regression model that isn't used in this course. 11. ExpReg - A regression model that isn't used in this course. 12. PwrReg - A regression model that isn't used in this course.

LIST Key The LIST command has two major sections, OPS (Operations) and MATH. LIST-OPS

1. SortA( - This will sort a list in ascending order. This command is equivalent to the SortA( command under the STATS key. 2. SortD( - This will sort a list in descending order. This command is equivalent to the SortD( command under the STATS key. 3. dim - This function will return the dimensions of a list. The dimension of a list the number of elements in the list. This is also used as a command to set the dimensions of a list. 4. Fill( - This command will fill a list with a constant. This is useful if you need to set an entire list to all be one number. 5. seq - This function will generate a sequence of numbers according to the function specified as the first argument. A list is returned, but you must save it to one of the six lists if you want to use it for anything. LIST-MATH

1. min( - Returns the minimum value in a list. 2. max( - Returns the maximum value in a list 3. mean( - Returns the arithmetic mean of all numbers in the list. The mean is the sum of the list divided by the dimension of the list. 4. median( - Returns the median of the list. The median is the middle number when the list is sorted in ascending order. If the dimension is an even number, the median is the midpoint between the two middle values when the list is sorted in ascending order. 5. sum - Returns the sum of the values in the list. 6. prod - Returns the product of the values in the list. If the product of a list is zero, then at least one of the numbers is zero.

Other Keys VARS

The VARS key can be used to retrieve the value of a statistic. VARS Statistics

This will save a lot of retyping of values and allow you to use the full accuracy of the calculator instead of losing digits when re-entering numbers. Here are some common values you will be using: Keystrokes

Statistic

VARS 5 1

n, the sample size

VARS 5 1

x bar, the sample mean

VARS 5 1

Sx, the sample standard deviation

VARS 5 1

minX, the minimum value

VARS 5 1

maxX, the maximum value

There are other values under statistics which you will use. You may have to arrow to other submenus first for some of them. STORE

This key will save values. You may save a scalar value to a real variable (A-Z) or a list value to a list (L1 - L6). You can use the STORE key to save a value to the dimension of a list to set its size. You can use the STORE key to save a list generated by the sequence command to a list. Mathematical Operations and Functions

Lists can be used as arguments of functions. If they are, the function is applied to each element in the list. Mathematical operations can be performed on lists. For more information on lists, see the Introduction to the TI-82.

Entering Data Always start with a clean set of data. You don't want to mix data from one problem with data from another problem. Before starting any new problem, you should clear out existing data. STAT ClrList L1,L2,L3

Another way to clear the lists is to go into STAT EDIT, arrow to the top so that the list name is highlighted. Then press the CLEAR key and ENTER. You may only need to specify one list, but you can specify more than one, just separate them with commas. After the lists have been cleared, you can enter the new lists: STAT Edit Select this list that you want to use. The default will be L1. This will be fine for most things, but do realize you can use any of the lists. Just be sure to check the setup later. Type in each number separating them by enter. When you are done entering, press the QUIT key (2nd MODE). If you need to correct data, just go back to STAT EDIT without clearing the list first.

TI-82: Histograms, BoxPlots You can use the calculator to draw histograms, box-plots, and compute the frequency of each class. See the instructions on using the calculator to do statistics and lists. This provides an overview as well as some helpful advice for working with statistics on the calculator.

Histograms 1. Enter the data. 2. Determine the class width and the lower class boundary (not limit) of the first class using the techniques for creating grouped frequency distributions. 3. Turn off any regular plots: Hit Y= and position the cursor over any equal sign which is in inversed video (white on black) by arrowing left and then down if necessary. Hit enter while the cursor is on the equal sign to toggle between displaying the function (equal sign highlighted) and not displaying the function (equal sign not highlighted). 4. Press the STATPLOT key (2nd Y=)

5. Select a plot (usually plot 1) and hit enter 6. Turn the plot on by highlighting the ON and pressing enter. 7. Set the TYPE to histograph (last type) 8. Set the XLIST to the list you put the data into 9. Set the FREQ to 1. 10. Select WINDOW 11. Put the lower class boundary for the first class in XMIN 12. The XMAX value should be the lower class boundary for the first class plus the number of classes times the class width. 13. The Class Width should be stored in XSCL 14. YMIN should be set to 0 15. YMAX should be at least the largest frequency in any class. This is difficult to know if you're generating the histogram without first writing the table by hand. If the histogram displayed doesn't fit on the screen, go back and change this number. A good initial guess might be the sample size divided by the number or classes. You might round up it to a nice number (multiple of 5) or add one or two so that graph is completely shown on the screen. 16. YSCL should be set based on the YMAX value. A factor of YMAX would be a good choice (so if YMAX is 30, let YSCL be 5). If your YMAX is small (say under 10), you might want to set it to 1. This will determine how many marks are placed along the vertical axis. 17. Hit the GRAPH key.

Finding the Frequency 1. 2. 3. 4. 5. 6.

Generate a histogram first Hit the TRACE key The "min" value is the lower class boundary The "max" value is the upper class boundary The "n" value is the frequency for that class. Use the left and right arrow keys to get the values for all the classes.

Box Plots 1. Enter the data. 2. Turn off any regular plots: Hit Y= and position the cursor over any equal sign which is in inversed video (white on black) by arrowing left and then down if necessary. Hit enter while the cursor is on the equal sign to toggle between displaying the function (equal sign highlighted) and not displaying the function (equal sign not highlighted).

3. 4. 5. 6. 7. 8. 9.

Press the STATPLOT key (2nd Y=) Select a plot (usually plot 1) and hit enter Turn the plot on by highlighting the ON and pressing enter. Set the TYPE to box-plot (3rd type) Set the XLIST to the list you put the data into Set the FREQ to 1. Zoom to Statistics mode (ZOOM 9)

You hit the TRACE key with the box plot displayed to find the five numbers associated with it. You may use the left and right arrow keys to find all five numbers. Note that the calculator uses the quartiles instead of the hinges. The hinges and quartiles are the same unless the remainder when the sample size is divided by four is three.

TI-82: Plotting an Ogive The Ogive is a frequency polygon (line plot) graph of the cumulative frequency or the relative cumulative frequency. The horizontal axis is marked with the class boundaries and the vertical axis is the frequency. All class boundaries are used -- there will be one more class boundary than the number of classes. The following example assumes the class boundaries are in List 1 and the cumulative frequencies are in List 2. You are free to use any two lists that you desire, but you should make the appropriate adjustments in the instructions if you don't use List 1 and List 2. 1. Enter the class boundaries into List 1. Start with the lower boundary of the first class and end with the upper boundary of the last class. 2. Enter the cumulative frequencies into List 2. Start with 0 for the first value because there is nothing less than the first lower class boundary. 3. Turn off any regular plots: Hit Y= and position the cursor over any equal sign which is in inversed video (white on black) by arrowing left and then down if necessary. Hit enter while the cursor is on the equal sign to toggle between displaying the function (equal sign highlighted) and not displaying the function (equal sign not highlighted). 4. Press the STATPLOT key (2nd Y=) 5. Select a plot (usually plot 1) and hit enter 6. Turn the plot on by highlighting the ON and pressing enter. 7. Set the TYPE to LinePlot (2nd type)

8. Set the XLIST to List 1 9. Set the YLIST to List 2 10. Set the MARKER to any of the three values 11. Select WINDOW 12. Put the lower class boundary for the first class in XMIN 13. The XMAX value should be the upper class boundary of the last class 14. The Class Width should be stored in XSCL 15. YMIN should be set to 0 16. YMAX should be set to the total frequency if using cumulative frequencies in List 2 and set to 1.00 if using relative cumulative frequencies in List 2. 17. YSCL should be set appropriately based on YMAX. 18. Hit the GRAPH key. Relative Frequencies

There is no need to re-enter the data if you wish to use relative cumulative frequencies instead of cumulative frequencies. The following assumes that the cumulative frequencies are in List 2. Replace the ### by the total frequency. You can't put "###" into the calculator. L2 / ### STORE L2

This will replace the cumulative frequencies with the relative cumulative frequencies. To replace relative cumulative frequencies with cumulative frequencies, change the division to multiplication. L2 * ### STORE L2

PIE Program The TI-82 doesn't pie charts directly as it does with scatterplots, box plots, and histograms. Place the frequencies or relative frequencies in List 1. If the List 1 is empty or the sum of list 1 is zero, then you are instructed to put the frequencies in list 1.

Turn off any graphs that may be on before running the PIE program. Otherwise, the graphs will overlay the pie chart and it will take longer to draw. The program will ask the if they wish to place the labels on the graph. If the enters 1 for yes, then the values in List 1 will be placed in the graph. This is where the difference between frequencies or relative frequencies appear. This program will force the calculator into radian mode and turn the axes off, zoom standard and then zoom square. It will then draw a circle and proceed to draw the lines which define the pie graph. To reset the graphing screen to normal when done viewing the pie chart, you need to: 1. DRAW CLRDRAW 2. WINDOW FORMAT AXESON 3. MODE DEGREE -Depending on your use, Leaving it in Radian mode may be preferred

Statistics: Data Description chp3 Definitions Statistic Characteristic or measure obtained from a sample Parameter Characteristic or measure obtained from a population Mean Sum of all the values divided by the number of values. This can either be a population mean (denoted by mu) or a sample mean (denoted by x bar) Median The midpoint of the data after being ranked (sorted in ascending order). There are as many numbers below the median as above the median.

Mode The most frequent number Skewed Distribution The majority of the values lie together on one side with a very few values (the tail) to the other side. In a positively skewed distribution, the tail is to the right and the mean is larger than the median. In a negatively skewed distribution, the tail is to the left and the mean is smaller than the median. Symmetric Distribution The data values are evenly distributed on both sides of the mean. In a symmetric distribution, the mean is the median. Weighted Mean The mean when each value is multiplied by its weight and summed. This sum is divided by the total of the weights. Midrange The mean of the highest and lowest values. (Max + Min) / 2 Range The difference between the highest and lowest values. Max - Min Population Variance The average of the squares of the distances from the population mean. It is the sum of the squares of the deviations from the mean divided by the population size. The units on the variance are the units of the population squared. Sample Variance Unbiased estimator of a population variance. Instead of dividing by the population size, the sum of the squares of the deviations from the sample mean is divided by one less than the sample size. The units on the variance are the units of the population squared. Standard Deviation

The square root of the variance. The population standard deviation is the square root of the population variance and the sample standard deviation is the square root of the sample variance. The sample standard deviation is not the unbiased estimator for the population standard deviation. The units on the standard deviation is the same as the units of the population/sample. Coefficient of Variation Standard deviation divided by the mean, expressed as a percentage. We won't work with the Coefficient of Variation in this course. Chebyshev's Theorem The proportion of the values that fall within k standard deviations of the mean is at least where k > 1. Chebyshev's theorem can be applied to any distribution regardless of its shape. Empirical or Normal Rule Only valid when a distribution in bell-shaped (normal). Approximately 68% lies within 1 standard deviation of the mean; 95% within 2 standard deviations; and 99.7% within 3 standard deviations of the mean. Standard Score or Z-Score The value obtained by subtracting the mean and dividing by the standard deviation. When all values are transformed to their standard scores, the new mean (for Z) will be zero and the standard deviation will be one. Percentile The percent of the population which lies below that value. The data must be ranked to find percentiles. Quartile Either the 25th, 50th, or 75th percentiles. The 50th percentile is also called the median. Decile Either the 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, or 90th percentiles.

Lower Hinge The median of the lower half of the numbers (up to and including the median). The lower hinge is the first Quartile unless the remainder when dividing the sample size by four is 3. Upper Hinge The median of the upper half of the numbers (including the median). The upper hinge is the 3rd Quartile unless the remainder when dividing the sample size by four is 3. Box and Whiskers Plot (Box Plot) A graphical representation of the minimum value, lower hinge, median, upper hinge, and maximum. Some textbooks, and the TI-82 calculator, define the five values as the minimum, first Quartile, median, third Quartile, and maximum. Five Number Summary Minimum value, lower hinge, median, upper hinge, and maximum. InterQuartile Range (IQR) The difference between the 3rd and 1st Quartiles. Outlier An extremely high or low value when compared to the rest of the values. Mild Outliers Values which lie between 1.5 and 3.0 times the InterQuartile Range below the 1st Quartile or above the 3rd Quartile. Note, some texts use hinges instead of Quartiles. Extreme Outliers Values which lie more than 3.0 times the InterQuartile Range below the 1st Quartile or above the 3rd Quartile. Note, some texts use hinges instead of Quartiles.

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Stats: Measures of Central Tendency The term "Average" is vague Average could mean one of four things. The arithmetic mean, the median, midrange, or mode. For this reason, it is better to specify which average you're talking about.

Mean This is what people usually intend when they say "average"

Population Mean:

Sample Mean:

Frequency Distribution: The mean of a frequency distribution is also the weighted mean.

Median The data must be ranked (sorted in ascending order) first. The median is the number in the middle. To find the depth of the median, there are several formulas that could be used, the one that we will use is: Depth of median = 0.5 * (n + 1)

Raw Data The median is the number in the "depth of the median" position. If the sample size is even, the depth of the median will be a decimal -- you need to find the midpoint between the numbers on either side of the depth of the median. Ungrouped Frequency Distribution Find the cumulative frequencies for the data. The first value with a cumulative frequency greater than depth of the median is the median. If the depth of the median is exactly 0.5 more than the cumulative frequency of the previous class, then the median is the midpoint between the two classes. Grouped Frequency Distribution This is the tough one. Since the data is grouped, you have lost all original information. Some textbooks have you simply take the midpoint of the class. This is an over-simplification which isn't the true value (but much easier to do). The correct process is to interpolate. Find out what proportion of the distance into the median class the median by dividing the sample size by 2, subtracting the cumulative frequency of the previous class, and then dividing all that bay the frequency of the median class. Multiply this proportion by the class width and add it to the lower boundary of the median class.

Mode The mode is the most frequent data value. There may be no mode if no one value appears more than any other. There may also be two modes (bimodal), three modes (trimodal), or more than three modes (multi-modal). For grouped frequency distributions, the modal class is the class with the largest frequency.

Midrange

The midrange is simply the midpoint between the highest and lowest values.

Summary The Mean is used in computing other statistics (such as the variance) and does not exist for open ended grouped frequency distributions (1). It is often not appropriate for skewed distributions such as salary information. The Median is the center number and is good for skewed distributions because it is resistant to change. The Mode is used to describe the most typical case. The mode can be used with nominal data whereas the others can't. The mode may or may not exist and there may be more than one value for the mode (2). The Midrange is not used very often. It is a very rough estimate of the average and is greatly affected by extreme values (even more so than the mean). Property

Mean

Median

Mode

Midrange

Always Exists

No (1)

Yes

No (2)

Yes

Uses all data values

Yes

No

No

No

Affected by extreme values

Yes

No

No

Yes

Using the TI-82 One can find the mean, median, and midrange using the list functions of the TI-82. You can also find the measures of variation with the TI-82 calculator.

Table of Contents

Stats: Measures of Variation

Range The range is the simplest measure of variation to find. It is simply the highest value minus the lowest value. RANGE = MAXIMUM - MINIMUM

Since the range only uses the largest and smallest values, it is greatly affected by extreme values, that is - it is not resistant to change.

Variance "Average Deviation" The range only involves the smallest and largest numbers, and it would be desirable to have a statistic which involved all of the data values. The first attempt one might make at this is something they might call the average deviation from the mean and define it as:

The problem is that this summation is always zero. So, the average deviation will always be zero. That is why the average deviation is never used. Population Variance So, to keep it from being zero, the deviation from the mean is squared and called the "squared deviation from the mean". This "average squared deviation from the mean" is called the variance.

Unbiased Estimate of the Population Variance One would expect the sample variance to simply be the population variance with the population mean replaced by the sample mean. However, one of the major uses of statistics is to estimate the corresponding parameter. This formula has the problem

that the estimated value isn't the same as the parameter. To counteract this, the sum of the squares of the deviations is divided by one less than the sample size.

Standard Deviation There is a problem with variances. Recall that the deviations were squared. That means that the units were also squared. To get the units back the same as the original data values, the square root must be taken.

The sample standard deviation is not the unbiased estimator for the population standard deviation. The calculator does not have a variance key on it. It does have a standard deviation key. You will have to square the standard deviation to find the variance.

Sum of Squares (shortcuts) The sum of the squares of the deviations from the means is given a shortcut notation and several alternative formulas.

A little algebraic simplification returns: What's wrong with the first formula, you ask? Consider the following example - the last row are the totals for the columns

1. 2. 3. 4.

Total the data values: 23 Divide by the number of values to get the mean: 23/5 = 4.6 Subtract the mean from each value to get the numbers in the second column. Square each number in the second column to get the values in the third column. 5. Total the numbers in the third column: 5.2 6. Divide this total by one less than the sample size to get the variance: 5.2 / 4 = 1.3 x 4

4 - 4.6 = -0.6

( - 0.6 )^2 = 0.36

5

5 - 4.6 = 0.4

( 0.4 ) ^2 = 0.16

3

3 - 4.6 = -1.6

( - 1.6 )^2 = 2.56

6

6 - 4.6 = 1.4

( 1.4 )^2 = 1.96

5

5 - 4.6 = 0.4

( 0.4 )^2 = 0.16

23

0.00 (Always)

5.2

Not too bad, you think. But this can get pretty bad if the sample mean doesn't happen to be an "nice" rational number. Think about having a mean of 19/7 = 2.714285714285... Those subtractions get nasty, and when you square them, they're really bad. Another problem with the first formula is that it requires you to know the mean ahead of time. For a calculator, this would mean that you have to save all of the numbers that were entered. The TI-82 does this, but most scientific calculators don't. Now, let's consider the shortcut formula. The only things that you need to find are the sum of the values and the sum of the values squared. There is no subtraction and no decimals or fractions until the end. The last row contains the sums of the columns, just like before. 1. Record each number in the first column and the square of each number in the second column. 2. Total the first column: 23 3. Total the second column: 111

4. Compute the sum of squares: 111 - 23*23/5 = 111 - 105.8 = 5.2 5. Divide the sum of squares by one less than the sample size to get the variance = 5.2 / 4 = 1.3 x

x^2

4

16

5

25

3

9

6

36

5

25

23

111

Chebyshev's Theorem The proportion of the values that fall within k standard deviations of the mean will be at least

, where k is an number greater than 1.

"Within k standard deviations" interprets as the interval:

to

.

Chebyshev's Theorem is true for any sample set, not matter what the distribution.

Empirical Rule The empirical rule is only valid for bell-shaped (normal) distributions. The following statements are true.   

Approximately 68% of the data values fall within one standard deviation of the mean. Approximately 95% of the data values fall within two standard deviations of the mean. Approximately 99.7% of the data values fall within three standard deviations of the mean.

The empirical rule will be revisited later in the chapter on normal probabilities.

Using the TI-82 to find these values You may use the TI-82 to find the measures of central tendency and the measures of variation using the list handling capabilities of the calculator.

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Stats: Measures of Position Standard Scores (z-scores) The standard score is obtained by subtracting the mean and dividing the difference by the standard deviation. The symbol is z, which is why it's also called a z-score.

The mean of the standard scores is zero and the standard deviation is 1. This is the nice feature of the standard score -- no matter what the original scale was, when the data is converted to its standard score, the mean is zero and the standard deviation is 1.

Percentiles, Deciles, Quartiles Percentiles (100 regions)

The kth percentile is the number which has k% of the values below it. The data must be ranked. 1. Rank the data 2. Find k% (k /100) of the sample size, n. 3. If this is an integer, add 0.5. If it isn't an integer round up.

4. Find the number in this position. If your depth ends in 0.5, then take the midpoint between the two numbers. It is sometimes easier to count from the high end rather than counting from the low end. For example, the 80th percentile is the number which has 80% below it and 20% above it. Rather than counting 80% from the bottom, count 20% from the top. Note: The 50th percentile is the median. If you wish to find the percentile for a number (rather than locating the kth percentile), then 1. 2. 3. 4.

Take the number of values below the number Add 0.5 Divide by the total number of values Convert it to a percent

Deciles (10 regions)

The percentiles divide the data into 100 equal regions. The deciles divide the data into 10 equal regions. The instructions are the same for finding a percentile, except instead of dividing by 100 in step 2, divide by 10. Quartiles (4 regions)

The quartiles divide the data into 4 equal regions. Instead of dividing by 100 in step 2, divide by 4. Note: The 2nd quartile is the same as the median. The 1st quartile is the 25th percentile, the 3rd quartile is the 75th percentile. The quartiles are commonly used (much more so than the percentiles or deciles). The TI-82 calculator will find the quartiles for you. Some textbooks include the quartiles in the five number summary.

Hinges The lower hinge is the median of the lower half of the data up to and including the median. The upper hinge is the median of the upper half of the data up to and including the median.

The hinges are the same as the quartiles unless the remainder when dividing the sample size by four is three (like 39 / 4 = 9 R 3). The statement about the lower half or upper half including the median tends to be confusing to some students. If the median is split between two values (which happens whenever the sample size is even), the median isn't included in either since the median isn't actually part of the data. Example 1: sample size of 20 The median will be in position 10.5. The lower half is positions 1 - 10 and the upper half is positions 11 - 20. The lower hinge is the median of the lower half and would be in position 5.5. The upper hinge is the median of the upper half and would be in position 5.5 starting with original position 11 as position 1 -- this is the original position 15.5. Example 2: sample size of 21 The median is in position 11. The lower half is positions 1 - 11 and the upper half is positions 11 - 21. The lower hinge is the median of the lower half and would be in position 6. The upper hinge is the median of the upper half and would be in position 6 when starting at position 11 -- this is original position 16.

Five Number Summary The five number summary consists of the minimum value, lower hinge, median, upper hinge, and maximum value. Some textbooks use the quartiles instead of the hinges.

Box and Whiskers Plot A graphical representation of the five number summary. A box is drawn between the lower and upper hinges with a line at the median. Whiskers (a single line, not a box) extend from the hinges to lines at the minimum and maximum values.

Interquartile Range (IQR) The interquartile range is the difference between the third and first quartiles. That's it: Q3 - Q1

Outliers

Outliers are extreme values. There are mild outliers and extreme outliers. The Bluman text does not distinguish between mild outliers and extreme outliers and just treats either as an outlier. Extreme Outliers

Extreme outliers are any data values which lie more than 3.0 times the interquartile range below the first quartile or above the third quartile. x is an extreme outlier if ... x < Q1 - 3 * IQR

or x > Q3 + 3 * IQR

Mild Outliers

Mild outliers are any data values which lie between 1.5 times and 3.0 times the interquartile range below the first quartile or above the third quartile. x is a mild outlier if ... Q1 - 3 * IQR <= x < Q1 - 1.5 * IQR

or Q1 + 1.5 * IQR < x <= Q3 + 3 * IQR

Stats: Counting Techniques chp4 Definitions Factorial A positive integer factorial is the product of each natural number up to and including the integer. Permutation An arrangement of objects in a specific order. Combination

A selection of objects without regard to order. Tree Diagram A graphical device used to list all possibilities of a sequence of events in a systematic way.

Table of Contents

Stats: Counting Techniques Fundamental Theorems Arithmetic Every integer greater than one is either prime or can be expressed as an unique product of prime numbers Algebra Every polynomial function on one variable of degree n > 0 has at least one real or complex zero. Linear Programming If there is a solution to a linear programming problem, then it will occur at a corner point or on a boundary between two or more corner points

Fundamental Counting Principle In a sequence of events, the total possible number of ways all events can performed is the product of the possible number of ways each individual event can be performed. The Bluman text calls this multiplication principle 2.

Factorials If n is a positive integer, then n! = n (n-1) (n-2) ... (3)(2)(1) n! = n (n-1)!

A special case is 0! 0! = 1

Permutations A permutation is an arrangement of objects without repetition where order is important. Permutations using all the objects A permutation of n objects, arranged into one group of size n, without repetition, and order being important is: nPn

= P(n,n) = n!

Example: Find all permutations of the letters "ABC" ABC

ACB

BAC

BCA

CAB

CBA

Permutations of some of the objects A permutation of n objects, arranged in groups of size r, without repetition, and order being important is: nPr

= P(n,r) = n! / (n-r)!

Example: Find all two-letter permutations of the letters "ABC" AB

AC

BA

BC

CA

CB

Shortcut formula for finding a permutation

Assuming that you start a n and count down to 1 in your factorials ... P(n,r) = first r factors of n factorial

Distinguishable Permutations Sometimes letters are repeated and all of the permutations aren't distinguishable from each other. Example: Find all permutations of the letters "BOB" To help you distinguish, I'll write the second "B" as "b" BOb

BbO

OBb

ObB

bBO

bOB

If you just write "B" as "B", however ... BOB

BBO

OBB

OBB

BBO

BBO

There are really only three distinguishable permutations here. BOB

BBO

OBB

If a word has N letters, k of which are unique, and you let n (n1, n2, n3, ..., nk) be the frequency of each of the k letters, then the total number of distinguishable permutations is given by:

Consider the word "STATISTICS": Here are the frequency of each letter: S=3, T=3, A=1, I=2, C=1, there are 10 letters total 10! 10*9*8*7*6*5*4*3*2*1 Permutations = -------------- = -------------------- = 50400 3! 3! 1! 2! 1! 6 * 6 * 1 * 2 * 1

You can find distinguishable permutations using the TI-82.

Combinations A combination is an arrangement of objects without repetition where order is not important.

Note: The difference between a permutation and a combination is not whether there is repetition or not -- there must not be repetition with either, and if there is repetition, you can not use the formulas for permutations or combinations. The only difference in the definition of a permutation and a combination is whether order is important. A combination of n objects, arranged in groups of size r, without repetition, and order being important is: nCr

= C(n,r) = n! / ( (n-r)! * r! )

Another way to write a combination of n things, r at a time is using the binomial notation: Example: Find all two-letter combinations of the letters "ABC" AB = BA

AC = CA

BC = CB

There are only three two-letter combinations. Shortcut formula for finding a combination

Assuming that you start a n and count down to 1 in your factorials ... C(n,r) = first r factors of n factorial divided by the last r factors of n factorial Pascal's Triangle Combinations are used in the binomial expansion theorem from algebra to give the coefficients of the expansion (a+b)^n. They also form a pattern known as Pascal's Triangle. 1 1 1 1 1

1 2

3

5

6

1

1 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1

4

1 3 4

10

Each element in the table is the sum of the two elements directly above it. Each element is also a combination. The n value is the number of the row (start counting at zero) and the r value is the element in the row (start counting at zero). That would

make the 20 in the next to last row C(6,3) -- it's in the row #6 (7th row) and position #3 (4th element). Symmetry Pascal's Triangle illustrates the symmetric nature of a combination. C(n,r)

= C(n,n-

r)

Example: C(10,4) = C(10,6) or C(100,99) = C(100,1) Shortcut formula for finding a combination

Since combinations are symmetric, if n-r is smaller than r, then switch the combination to its alternative form and then use the shortcut given above. C(n,r) = first r factors of n factorial divided by the last r factors of n factorial

TI-82 You can use the TI-82 graphing calculator to find factorials, permutations, and combinations.

Tree Diagrams Tree diagrams are a graphical way of listing all the possible outcomes. The outcomes are listed in an orderly fashion, so listing all of the possible outcomes is easier than just trying to make sure that you have them all listed. It is called a tree diagram because of the way it looks. The first event appears on the left, and then each sequential event is represented as branches off of the first event. The tree diagram to the right would show the possible ways of flipping two coins. The final outcomes are obtained by following each branch to its conclusion: They are from top to bottom: HH

HT

TH

TT

Table of Contents

Stats: Probability chp5 Definitions Probability Experiment Process which leads to well-defined results call outcomes Outcome The result of a single trial of a probability experiment Sample Space Set of all possible outcomes of a probability experiment Event One or more outcomes of a probability experiment Classical Probability Uses the sample space to determine the numerical probability that an event will happen. Also called theoretical probability. Equally Likely Events Events which have the same probability of occurring. Complement of an Event All the events in the sample space except the given events. Empirical Probability Uses a frequency distribution to determine the numerical probability. An empirical probability is a relative frequency. Subjective Probability

Uses probability values based on an educated guess or estimate. It employs opinions and inexact information. Mutually Exclusive Events Two events which cannot happen at the same time. Dist Events Another name for mutually exclusive events. Independent Events Two events are independent if the occurrence of one does not affect the probability of the other occurring. Dependent Events Two events are dependent if the first event affects the outcome or occurrence of the second event in a way the probability is changed. Conditional Probability The probability of an event occurring given that another event has already occurred. Bayes' Theorem A formula which allows one to find the probability that an event occurred as the result of a particular previous event.

Table of Contents

Stats: Introduction to Probability Sample Spaces

A sample space is the set of all possible outcomes. However, some sample spaces are better than others. Consider the experiment of flipping two coins. It is possible to get 0 heads, 1 head, or 2 heads. Thus, the sample space could be {0, 1, 2}. Another way to look at it is flip { HH, HT, TH, TT }. The second way is better because each event is as equally likely to occur as any other. When writing the sample space, it is highly desirable to have events which are equally likely. Another example is rolling two dice. The sums are { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }. However, each of these aren't equally likely. The only way to get a sum 2 is to roll a 1 on both dice, but you can get a sum of 4 by rolling a 1-3, 2-2, or 3-1. The following table illustrates a better sample space for the sum obtain when rolling two dice. Second Die First Die

1

2

3

4

5

6

1

2

3

4

5

6

7

2

3

4

5

6

7

8

3

4

5

6

7

8

9

4

5

6

7

8

9

10

5

6

7

8

9

10

11

6

7

8

9

10

11

12

Classical Probability The above table lends itself to describing data another way -- using a probability distribution. Let's consider the frequency distribution for the above sums. Sum

Frequency

Relative Frequency

2

1

1/36

3

2

2/36

4

3

3/36

5

4

4/36

6

5

5/36

7

6

6/36

8

5

5/36

9

4

4/36

10

3

3/36

11

2

2/36

12

1

1/36

If just the first and last columns were written, we would have a probability distribution. The relative frequency of a frequency distribution is the probability of the event occurring. This is only true, however, if the events are equally likely. This gives us the formula for classical probability. The probability of an event occurring is the number in the event divided by the number in the sample space. Again, this is only true when the events are equally likely. A classical probability is the relative frequency of each event in the sample space when each event is equally likely.

P(E) = n(E) / n(S) Empirical Probability Empirical probability is based on observation. The empirical probability of an event is the relative frequency of a frequency distribution based upon observation.

P(E) = f / n

Probability Rules There are two rules which are very important. All probabilities are between 0 and 1 inclusive

0 <= P(E) <= 1 The sum of all the probabilities in the sample space is 1 There are some other rules which are also important. The probability of an event which cannot occur is 0. The probability of any event which is not in the sample space is zero. The probability of an event which must occur is 1. The probability of the sample space is 1. The probability of an event not occurring is one minus the probability of it occurring. P(E') = 1 - P(E) Continue and learn more about the rules of probability.

Table of Contents

Stats: Probability Rules "OR" or Unions Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is dist. If two events are dist, then the probability of them both occurring at the same time is 0.

Dist:

P(A and B) = 0

If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring. Specific Addition Rule Only valid when the events are mutually exclusive. P(A or B) = P(A) + P(B) Example 1: Given: P(A) = 0.20, P(B) = 0.70, A and B are dist I like to use what's called a t probability distribution. (Since dist means nothing in common, t is what they have in common -- so the values that go on the inside portion of the table are the intersections or "and"s of each pair of events). "Marginal" is another word for totals -- it's called marginal because they appear in the margins. B

B'

Marginal

A

0.00

0.20

0.20

A'

0.70

0.10

0.80

Marginal

0.70

0.30

1.00

The values in red are given in the problem. The grand total is always 1.00. The rest of the values are obtained by addition and subtraction. Non-Mutually Exclusive Events

In events which aren't mutually exclusive, there is some overlap. When P(A) and P(B) are added, the probability of the intersection (and) is added twice. To compensate for that double addition, the intersection needs to be subtracted. General Addition Rule Always valid.

P(A or B) = P(A) + P(B) - P(A and B) Example 2: Given P(A) = 0.20, P(B) = 0.70, P(A and B) = 0.15 B

B'

Marginal

A

0.15

0.05

0.20

A'

0.55

0.25

0.80

Marginal

0.70

0.30

1.00

Interpreting the table

Certain things can be determined from the t probability distribution. Mutually exclusive events will have a probability of zero. All inclusive events will have a zero opposite the intersection. All inclusive means that there is nothing outside of those two events: P(A or B) = 1. B

B'

Marginal

A

A and B are Mutually Exclusive if this value is 0

.

.

A'

.

A and B are All Inclusive if this value is 0

.

Marginal

.

.

1.00

"AND" or Intersections Independent Events

Two events are independent if the occurrence of one does not change the probability of the other occurring. An example would be rolling a 2 on a die and flipping a head on a coin. Rolling the 2 does not affect the probability of flipping the head.

If events are independent, then the probability of them both occurring is the product of the probabilities of each occurring. Specific Multiplication Rule Only valid for independent events P(A and B) = P(A) * P(B) Example 3: P(A) = 0.20, P(B) = 0.70, A and B are independent. B

B'

Marginal

A

0.14

0.06

0.20

A'

0.56

0.24

0.80

Marginal

0.70

0.30

1.00

The 0.14 is because the probability of A and B is the probability of A times the probability of B or 0.20 * 0.70 = 0.14. Dependent Events

If the occurrence of one event does affect the probability of the other occurring, then the events are dependent. Conditional Probability The probability of event B occurring that event A has already occurred is read "the probability of B given A" and is written: P(B|A) General Multiplication Rule Always works. P(A and B) = P(A) * P(B|A)

Example 4: P(A) = 0.20, P(B) = 0.70, P(B|A) = 0.40 A good way to think of P(B|A) is that 40% of A is B. 40% of the 20% which was in event A is 8%, thus the intersection is 0.08. B

B'

Marginal

A

0.08

0.12

0.20

A'

0.62

0.18

0.80

Marginal

0.70

0.30

1.00

Independence Revisited

The following four statements are equivalent 1. 2. 3. 4.

A and B are independent events P(A and B) = P(A) * P(B) P(A|B) = P(A) P(B|A) = P(B)

The last two are because if two events are independent, the occurrence of one doesn't change the probability of the occurrence of the other. This means that the probability of B occurring, whether A has happened or not, is simply the probability of B occurring. Continue with conditional probabilities.

Table of Contents

Stats: Conditional Probability Conditional Probability

Recall that the probability of an event occurring given that another event has already occurred is called a conditional probability. The probability that event B occurs, given that event A has already occurred is P(B|A) = P(A and B) / P(A) This formula comes from the general multiplication principle and a little bit of algebra. Since we are given that event A has occurred, we have a reduced sample space. Instead of the entire sample space S, we now have a sample space of A since we know A has occurred. So the old rule about being the number in the event divided by the number in the sample space still applies. It is the number in A and B (must be in A since A has occurred) divided by the number in A. If you then divided numerator and denominator of the right hand side by the number in the sample space S, then you have the probability of A and B divided by the probability of A. Examples

Example 1: The question, "Do you smoke?" was asked of 100 people. Results are shown in the table.





.

Yes

No

Total

Male

19

41

60

Female

12

28

40

Total

31

69

100

What is the probability of a randomly selected individual being a male who smokes? This is just a t probability. The number of "Male and Smoke" divided by the total = 19/100 = 0.19 What is the probability of a randomly selected individual being a male? This is the total for male divided by the total = 60/100 = 0.60. Since no mention is made of smoking or not smoking, it includes all the cases.







What is the probability of a randomly selected individual smoking? Again, since no mention is made of gender, this is a marginal probability, the total who smoke divided by the total = 31/100 = 0.31. What is the probability of a randomly selected male smoking? This time, you're told that you have a male - think of stratified sampling. What is the probability that the male smokes? Well, 19 males smoke out of 60 males, so 19/60 = 0.31666... What is the probability that a randomly selected smoker is male? This time, you're told that you have a smoker and asked to find the probability that the smoker is also male. There are 19 male smokers out of 31 total smokers, so 19/31 = 0.6129 (approx)

After that last part, you have just worked a Bayes' Theorem problem. I know you didn't realize it - that's the beauty of it. A Bayes' problem can be set up so it appears to be just another conditional probability. In this class we will treat Bayes' problems as another conditional probability and not involve the large messy formula given in the text (and every other text). Example 2: There are three major manufacturing companies that make a product: Aberations, Brochmailians, and Chompielians. Aberations has a 50% market share, and Brochmailians has a 30% market share. 5% of Aberations' product is defective, 7% of Brochmailians' product is defective, and 10% of Chompieliens' product is defective. This information can be placed into a t probability distribution Company

Good

Defective

Total

Aberations

0.50-0.025 = 0.475

0.05(0.50) = 0.025

0.50

Brochmailians

0.30-0.021 = 0.279

0.07(0.30) = 0.021

0.30

Chompieliens

0.20-0.020 = 0.180

0.10(0.20) = 0.020

0.20

Total

0.934

0.066

The percent of the market share for Chompieliens wasn't given, but since the marginals must add to be 1.00, they have a 20% market share.

1.00

Notice that the 5%, 7%, and 10% defective rates don't go into the table directly. This is because they are conditional probabilities and the table is a t probability table. These defective probabilities are conditional upon which company was given. That is, the 7% is not P(Defective), but P(Defective|Brochmailians). The t probability P(Defective and Brochmailians) = P(Defective|Brochmailians) * P(Brochmailians). The "good" probabilities can be found by subtraction as shown above, or by multiplication using conditional probabilities. If 7% of Brochmailians' product is defective, then 93% is good. 0.93(0.30)=0.279.  



What is the probability a randomly selected product is defective? P(Defective) = 0.066 What is the probability that a defective product came from Brochmailians? P(Brochmailian|Defective) = P(Brochmailian and Defective) / P(Defective) = 0.021/0.066 = 7/22 = 0.318 (approx). Are these events independent? No. If they were, then P(Brochmailians|Defective)=0.318 would have to equal the P(Brochmailians)=0.30, but it doesn't. Also, the P(Aberations and Defective)=0.025 would have to be P(Aberations)*P(Defective) = 0.50*0.066=0.033, and it doesn't.

The second question asked above is a Bayes' problem. Again, my point is, you don't have to know Bayes formula just to work a Bayes' problem.

Bayes' Theorem However, just for the sake of argument, let's say that you want to know what Bayes' formula is. Let's use the same example, but shorten each event to its one letter initial, ie: A, B, C, and D instead of Aberations, Brochmailians, Chompieliens, and Defective. P(D|B) is not a Bayes problem. This is given in the problem. Bayes' formula finds the reverse conditional probability P(B|D). It is based that the Given (D) is made of three parts, the part of D in A, the part of D in B, and the part of D in C. P(B|D) =

P(B and D) ----------------------------------------P(A and D) + P(B and D) + P(C and D)

Inserting the multiplication rule for each of these t probabilities gives P(B|D) =

P(D|B)*P(B) ----------------------------------------P(D|A)*P(A) + P(D|B)*P(B) + P(D|C)*P(C)

However, and I hope you agree, it is much easier to take the t probability divided by the marginal probability. The table does the adding for you and makes the problems doable without having to memorize the formulas.

Table of Contents

Stats: Probability Distributions chp 6 Definitions Random Variable Variable whose values are determined by chance Probability Distribution The values a random variable can assume and the corresponding probabilities of each. Expected Value The theoretical mean of the variable. Binomial Experiment An experiment with a fixed number of independent trials. Each trial can only have two outcomes, or outcomes which can be reduced to two outcomes. The probability of each outcome must remain constant from trial to trial. Binomial Distribution The outcomes of a binomial experiment with their corresponding probabilities.

Multinomial Distribution A probability distribution resulting from an experiment with a fixed number of independent trials. Each trial has two or more mutually exclusive outcomes. The probability of each outcome must remain constant from trial to trial. Poisson Distribution A probability distribution used when a density of items is distributed over a period of time. The sample size needs to be large and the probability of success to be small. Hypergeometric Distribution A probability distribution of a variable with two outcomes when sampling is done without replacement.

Stats: Probability Distributions Probability Functions A probability function is a function which assigns probabilities to the values of a random variable.  

All the probabilities must be between 0 and 1 inclusive The sum of the probabilities of the outcomes must be 1.

If these two conditions aren't met, then the function isn't a probability function. There is no requirement that the values of the random variable only be between 0 and 1, only that the probabilities be between 0 and 1.

Probability Distributions A listing of all the values the random variable can assume with their corresponding probabilities make a probability distribution. A note about random variables. A random variable does not mean that the values can be anything (a random number). Random variables have a well defined set of outcomes and well defined probabilities for the occurrence of each outcome. The

random refers to the fact that the outcomes happen by chance -- that is, you don't know which outcome will occur next. Here's an example probability distribution that results from the rolling of a single fair die. x

1

2

3

4

5

6

sum

p(x)

1/6

1/6

1/6

1/6

1/6

1/6

6/6=1

Mean, Variance, and Standard Deviation Consider the following. The definitions for population mean and variance used with an ungrouped frequency distribution were: Some of you might be confused by only dividing by N. Recall that this is the population variance, the sample variance, which was the unbiased estimator for the population variance was when it was divided by n-1.

Using algebra, this is equivalent to: Recall that a probability is a long term relative frequency. So every f/N can be replaced by p(x). This simplifies to be: What's even better, is that the last portion of the variance is the mean squared. So, the two formulas that we will be using are:

Here's the example we were working on earlier. x

1

2

3

4

5

6

sum

p(x)

1/6

1/6

1/6

1/6

1/6

1/6

6/6 = 1

x p(x)

1/6

2/6

3/6

4/6

5/6

6/6

21/6 = 3.5

x^2 p(x)

1/6

4/6

9/6

16/6

25/6

36/6

91/6 = 15.1667

The mean is 7/2 or 3.5 The variance is 91/6 - (7/2)^2 = 35/12 = 2.916666... The standard deviation is the square root of the variance = 1.7078 Do not use rounded off values in the intermediate calculations. Only round off the final answer. You can learn how to find the mean and variance of a probability distribution using lists with the TI-82 or using the program called pdist.

Stats: Binomial Probabilities

Binomial Experiment A binomial experiment is an experiment which satisfies these four conditions    

A fixed number of trials Each trial is independent of the others There are only two outcomes The probability of each outcome remains constant from trial to trial.

These can be summarized as: An experiment with a fixed number of independent trials, each of which can only have two possible outcomes. The fact that each trial is independent actually means that the probabilities remain constant. Examples of binomial experiments  

Tossing a coin 20 times to see how many tails occur. Asking 200 people if they watch ABC news.



Rolling a die to see if a 5 appears.

Examples which aren't binomial experiments   

Rolling a die until a 6 appears (not a fixed number of trials) Asking 20 people how old they are (not two outcomes) Drawing 5 cards from a deck for a poker hand (done without replacement, so not independent)

Binomial Probability Function Example: What is the probability of rolling exactly two sixes in 6 rolls of a die? There are five things you need to do to work a binomial story problem. 1. Define Success first. Success must be for a single trial. Success = "Rolling a 6 on a single die" 2. Define the probability of success (p): p = 1/6 3. Find the probability of failure: q = 5/6 4. Define the number of trials: n = 6 5. Define the number of successes out of those trials: x = 2 Anytime a six appears, it is a success (denoted S) and anytime something else appears, it is a failure (denoted F). The ways you can get exactly 2 successes in 6 trials are given below. The probability of each is written to the right of the way it could occur. Because the trials are independent, the probability of the event (all six dice) is the product of each probability of each outcome (die) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

FFFFSS FFFSFS FFFSSF FFSFFS FFSFSF FFSSFF FSFFFS FSFFSF FSFSFF FSSFFF SFFFFS SFFFSF SFFSFF SFSFFF SSFFFF

5/6 5/6 5/6 5/6 5/6 5/6 5/6 5/6 5/6 5/6 1/6 1/6 1/6 1/6 1/6

* * * * * * * * * * * * * * *

5/6 5/6 5/6 5/6 5/6 5/6 1/6 1/6 1/6 1/6 5/6 5/6 5/6 5/6 1/6

* * * * * * * * * * * * * * *

5/6 5/6 5/6 1/6 1/6 1/6 5/6 5/6 5/6 1/6 5/6 5/6 5/6 1/6 5/6

* * * * * * * * * * * * * * *

5/6 1/6 1/6 5/6 5/6 1/6 5/6 5/6 1/6 5/6 5/6 5/6 1/6 5/6 5/6

* * * * * * * * * * * * * * *

1/6 5/6 1/6 5/6 1/6 5/6 5/6 1/6 5/6 5/6 5/6 1/6 5/6 5/6 5/6

* * * * * * * * * * * * * * *

1/6 1/6 5/6 1/6 5/6 5/6 1/6 5/6 5/6 5/6 1/6 5/6 5/6 5/6 5/6

= = = = = = = = = = = = = = =

(1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2 (1/6)^2

* * * * * * * * * * * * * * *

(5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4 (5/6)^4

Notice that each of the 15 probabilities are exactly the same: (1/6)^2 * (5/6)^4. Also, note that the 1/6 is the probability of success and you needed 2 successes. The 5/6 is the probability of failure, and if 2 of the 6 trials were success, then 4 of the 6 must be failures. Note that 2 is the value of x and 4 is the value of n-x. Further note that there are fifteen ways this can occur. This is the number of ways 2 successes can be occur in 6 trials without repetition and order not being important, or a combination of 6 things, 2 at a time. The probability of getting exactly x success in n trials, with the probability of success on a single trial being p is: P(X=x) = nCx * p^x * q^(n-x) Example: A coin is tossed 10 times. What is the probability that exactly 6 heads will occur. 1. 2. 3. 4. 5.

Success = "A head is flipped on a single coin" p = 0.5 q = 0.5 n = 10 x=6

P(x=6) = 10C6 * 0.5^6 * 0.5^4 = 210 * 0.015625 * 0.0625 = 0.205078125

Mean, Variance, and Standard Deviation The mean, variance, and standard deviation of a binomial distribution are extremely easy to find.

Another way to the variance is mu-q (since the np is mu).

Example: Find the mean, variance, and standard deviation for the number of sixes that appear when rolling 30 dice. Success = "a six is rolled on a single die". p = 1/6, q = 5/6. The mean is 30 * (1/6) = 5. The variance is 30 * (1/6) * (5/6) = 25/6. The standard deviation is the square root of the variance = 2.041241452 (approx)

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Stats: Other Discrete Distributions

Multinomial Probabilities A multinomial experiment is an extended binomial probability. The difference is that in a multinomial experiment, there are more than two possible outcomes. However, there are still a fixed number of independent trials, and the probability of each outcome must remain constant from trial to trial. Instead of using a combination, as in the case of the binomial probability, the number of ways the outcomes can occur is done using distinguishable permutations. An example here will be much more useful than a formula. The probability that a person will a College Algebra class is 0.55, the probability that a person will withdraw before the class is completed is 0.40, and the probability that a person will fail the class is 0.05. Find the probability that in a class of 30 students, exactly 16 , 12 withdraw, and 2 fail.

Outcome

x

p(outcome)



16

0.55

Withdraw

12

0.40

Fail

2

0.05

Total

30

1.00

The probability is found using this formula: 30! P = ---------------- * 0.55^16 * 0.40^12 * 0.05^2 (16!) (12!) (2!)

You can do this on the TI-82.

Poisson Probabilities Named after the French mathematician Simeon Poisson, Poisson probabilities are useful when there are a large number of independent trials with a small probability of success on a single trial and the variables occur over a period of time. It can also be used when a density of items is distributed over a given area or volume.

Lambda in the formula is the mean number of occurrences. If you're approximating a binomial probability using the Poisson, then lambda is the same as mu or n * p. Example: If there are 500 customers per eight-hour day in a check-out lane, what is the probability that there will be exactly 3 in line during any five-minute period? The expected value during any one five minute period would be 500 / 96 = 5.2083333. The 96 is because there are 96 five-minute periods in eight hours. So, you expect about 5.2 customers in 5 minutes and want to know the probability of getting exactly 3.

p(3;500/96) = e^(-500/96) * (500/96)^3 / 3! = 0.1288 (approx)

Hypergeometric Probabilities Hypergeometric experiments occur when the trials are not independent of each other and occur due to sampling without replacement -- as in a five card poker hand. Hypergeometric probabilities involve the multiplication of two combinations together and then division by the total number of combinations. Example: How many ways can 3 men and 4 women be selected from a group of 7 men and 10 women?

The answer is

= 7350/19448 = 0.3779 (approx)

Note that the sum of the numbers in the numerator are the numbers used in the combination in the denominator. This can be extended to more than two groups and called an extended hypergeometric problem. You can use the TI-82 to find hypergeometric probabilities.

Stats: Normal Distribution chp7 Definitions Central Limit Theorem Theorem which stats as the sample size increases, the sampling distribution of the sample means will become approximately normally distributed.

Correction for Continuity A correction applied to convert a discrete distribution to a continuous distribution. Finite Population Correction Factor A correction applied to the standard error of the means when the sample size is more than 5% of the population size and the sampling is done without replacement. Sampling Distribution of the Sample Means Distribution obtained by using the means computed from random samples of a specific size. Sampling Error Difference which occurs between the sample statistic and the population parameter due to the fact that the sample isn't a perfect representation of the population. Standard Error or the Mean The standard deviation of the sampling distribution of the sample means. It is equal to the standard deviation of the population divided by the square root of the sample size. Standard Normal Distribution A normal distribution in which the mean is 0 and the standard deviation is 1. It is denoted by z. Z-score Also known as z-value. A standardized score in which the mean is zero and the standard deviation is 1. The Z score is used to represent the standard normal distribution.

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Stats - Normal Distributions Any Normal Distribution      





Bell-shaped Symmetric about mean Continuous Never touches the x-axis Total area under curve is 1.00 Approximately 68% lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean. This is the Empirical Rule mentioned earlier. Data values represented by x which has mean mu and standard deviation sigma.

Probability Function given by

Standard Normal Distribution Same as a normal distribution, but also ...



Mean is zero Variance is one Standard Deviation is one Data values represented by z.



Probability Function given by

  

Normal Probabilities

Comprehension of this table is vital to success in the course!

There is a table which must be used to look up standard normal probabilities. The zscore is broken into two parts, the whole number and tenth are looked up along the left side and the hundredth is looked up across the top. The value in the intersection of the row and column is the area under the curve between zero and the z-score looked up. Because of the symmetry of the normal distribution, look up the absolute value of any z-score. Computing Normal Probabilities

There are several different situations that can arise when asked to find normal probabilities. Situation

Instructions

Between zero and any number

Look up the area in the table

Between two positives, or Between two negatives

Look up both areas in the table and subtract the smaller from the larger.

Between a negative and a positive

Look up both areas in the table and add them together

Less than a negative, or Greater than a positive

Look up the area in the table and subtract from 0.5000

Greater than a negative, or Less than a positive

Look up the area in the table and add to 0.5000

This can be shortened into two rules. 1. If there is only one z-score given, use 0.5000 for the second area, otherwise look up both z-scores in the table 2. If the two numbers are the same sign, then subtract; if they are different signs, then add. If there is only one z-score, then use the inequality to determine the second sign (< is negative, and > is positive).

Finding z-scores from probabilities

This is more difficult, and requires you to use the table inversely. You must look up the area between zero and the value on the inside part of the table, and then read the zscore from the outside. Finally, decide if the z-score should be positive or negative, based on whether it was on the left side or the right side of the mean. , zscores can be negative, but areas or probabilities cannot be. Situation

Instructions

Area between 0 and a value

Look up the area in the table Make negative if on the left side

Area in one tail

Subtract the area from 0.5000 Look up the difference in the table Make negative if in the left tail

Area including one complete half (Less than a positive or greater than a negative)

Subtract 0.5000 from the area Look up the difference in the table Make negative if on the left side

Within z units of the mean

Divide the area by 2 Look up the quotient in the table Use both the positive and negative z-scores

Two tails with equal area (More than z units from the mean)

Subtract the area from 1.000 Divide the area by 2 Look up the quotient in the table Use both the positive and negative z-scores

Using the table becomes proficient with practice, work lots of the normal probability problems!

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Standard Normal Probabilities

z

0.0 0

0.0 1

0.0 2

0.0 3

0.0 4

0.0 5

0.0 6

0.0 7

0.0 8

0.0 9

0 . 0

0.0 00 0

0.0 04 0

0.0 08 0

0.0 12 0

0.0 16 0

0.0 19 9

0.0 23 9

0.0 27 9

0.0 31 9

0.0 35 9

0 . 1

0.0 39 8

0.0 43 8

0.0 47 8

0.0 51 7

0.0 55 7

0.0 59 6

0.0 63 6

0.0 67 5

0.0 71 4

0.0 75 3

0 . 2

0.0 79 3

0.0 83 2

0.0 87 1

0.0 91 0

0.0 94 8

0.0 98 7

0.1 02 6

0.1 06 4

0.1 10 3

0.1 14 1

0 . 3

0.1 17 9

0.1 21 7

0.1 25 5

0.1 29 3

0.1 33 1

0.1 36 8

0.1 40 6

0.1 44 3

0.1 48 0

0.1 51 7

0 . 4

0.1 55 4

0.1 59 1

0.1 62 8

0.1 66 4

0.1 70 0

0.1 73 6

0.1 77 2

0.1 80 8

0.1 84 4

0.1 87 9

0 . 5

0.1 91 5

0.1 95 0

0.1 98 5

0.2 01 9

0.2 05 4

0.2 08 8

0.2 12 3

0.2 15 7

0.2 19 0

0.2 22 4

0 . 6

0.2 25 7

0.2 29 1

0.2 32 4

0.2 35 7

0.2 38 9

0.2 42 2

0.2 45 4

0.2 48 6

0.2 51 7

0.2 54 9

0 . 7

0.2 58 0

0.2 61 1

0.2 64 2

0.2 67 3

0.2 70 4

0.2 73 4

0.2 76 4

0.2 79 4

0.2 82 3

0.2 85 2

0 . 8

0.2 88 1

0.2 91 0

0.2 93 9

0.2 96 7

0.2 99 5

0.3 02 3

0.3 05 1

0.3 07 8

0.3 10 6

0.3 13 3

z

0.0 0

0.0 1

0.0 2

0.0 3

0.0 4

0.0 5

0.0 6

0.0 7

0.0 8

0.0 9

0 . 9

0.3 15 9

0.3 18 6

0.3 21 2

0.3 23 8

0.3 26 4

0.3 28 9

0.3 31 5

0.3 34 0

0.3 36 5

0.3 38 9

1 . 0

0.3 41 3

0.3 43 8

0.3 46 1

0.3 48 5

0.3 50 8

0.3 53 1

0.3 55 4

0.3 57 7

0.3 59 9

0.3 62 1

1 . 1

0.3 64 3

0.3 66 5

0.3 68 6

0.3 70 8

0.3 72 9

0.3 74 9

0.3 77 0

0.3 79 0

0.3 81 0

0.3 83 0

1 . 2

0.3 84 9

0.3 86 9

0.3 88 8

0.3 90 7

0.3 92 5

0.3 94 4

0.3 96 2

0.3 98 0

0.3 99 7

0.4 01 5

1 . 3

0.4 03 2

0.4 04 9

0.4 06 6

0.4 08 2

0.4 09 9

0.4 11 5

0.4 13 1

0.4 14 7

0.4 16 2

0.4 17 7

1 . 4

0.4 19 2

0.4 20 7

0.4 22 2

0.4 23 6

0.4 25 1

0.4 26 5

0.4 27 9

0.4 29 2

0.4 30 6

0.4 31 9

1 . 5

0.4 33 2

0.4 34 5

0.4 35 7

0.4 37 0

0.4 38 2

0.4 39 4

0.4 40 6

0.4 41 8

0.4 42 9

0.4 44 1

1 . 6

0.4 45 2

0.4 46 3

0.4 47 4

0.4 48 4

0.4 49 5

0.4 50 5

0.4 51 5

0.4 52 5

0.4 53 5

0.4 54 5

1 . 7

0.4 55 4

0.4 56 4

0.4 57 3

0.4 58 2

0.4 59 1

0.4 59 9

0.4 60 8

0.4 61 6

0.4 62 5

0.4 63 3

z

0.0 0

0.0 1

0.0 2

0.0 3

0.0 4

0.0 5

0.0 6

0.0 7

0.0 8

0.0 9

1 . 8

0.4 64 1

0.4 64 9

0.4 65 6

0.4 66 4

0.4 67 1

0.4 67 8

0.4 68 6

0.4 69 3

0.4 69 9

0.4 70 6

1 . 9

0.4 71 3

0.4 71 9

0.4 72 6

0.4 73 2

0.4 73 8

0.4 74 4

0.4 75 0

0.4 75 6

0.4 76 1

0.4 76 7

2 . 0

0.4 77 2

0.4 77 8

0.4 78 3

0.4 78 8

0.4 79 3

0.4 79 8

0.4 80 3

0.4 80 8

0.4 81 2

0.4 81 7

2 . 1

0.4 82 1

0.4 82 6

0.4 83 0

0.4 83 4

0.4 83 8

0.4 84 2

0.4 84 6

0.4 85 0

0.4 85 4

0.4 85 7

2 . 2

0.4 86 1

0.4 86 4

0.4 86 8

0.4 87 1

0.4 87 5

0.4 87 8

0.4 88 1

0.4 88 4

0.4 88 7

0.4 89 0

2 . 3

0.4 89 3

0.4 89 6

0.4 89 8

0.4 90 1

0.4 90 4

0.4 90 6

0.4 90 9

0.4 91 1

0.4 91 3

0.4 91 6

2 . 4

0.4 91 8

0.4 92 0

0.4 92 2

0.4 92 5

0.4 92 7

0.4 92 9

0.4 93 1

0.4 93 2

0.4 93 4

0.4 93 6

2 . 5

0.4 93 8

0.4 94 0

0.4 94 1

0.4 94 3

0.4 94 5

0.4 94 6

0.4 94 8

0.4 94 9

0.4 95 1

0.4 95 2

2 . 6

0.4 95 3

0.4 95 5

0.4 95 6

0.4 95 7

0.4 95 9

0.4 96 0

0.4 96 1

0.4 96 2

0.4 96 3

0.4 96 4

z

0.0 0

0.0 1

0.0 2

0.0 3

0.0 4

0.0 5

0.0 6

0.0 7

0.0 8

0.0 9

2 . 7

0.4 96 5

0.4 96 6

0.4 96 7

0.4 96 8

0.4 96 9

0.4 97 0

0.4 97 1

0.4 97 2

0.4 97 3

0.4 97 4

2 . 8

0.4 97 4

0.4 97 5

0.4 97 6

0.4 97 7

0.4 97 7

0.4 97 8

0.4 97 9

0.4 97 9

0.4 98 0

0.4 98 1

2 . 9

0.4 98 1

0.4 98 2

0.4 98 2

0.4 98 3

0.4 98 4

0.4 98 4

0.4 98 5

0.4 98 5

0.4 98 6

0.4 98 6

3 . 0

0.4 98 7

0.4 98 7

0.4 98 7

0.4 98 8

0.4 98 8

0.4 98 9

0.4 98 9

0.4 98 9

0.4 99 0

0.4 99 0

The values in the table are the areas between zero and the z-score. That is, P(0

Stats: Central Limit Theorem Sampling Distribution of the Sample Means Instead of working with individual scores, statisticians often work with means. What happens is that several samples are taken, the mean is computed for each sample, and then the means are used as the data, rather than individual scores being used. The sample is a sampling distribution of the sample means. When all of the possible sample means are computed, then the following properties are true:  

The mean of the sample means will be the mean of the population The variance of the sample means will be the variance of the population divided by the sample size.



 

The standard deviation of the sample means (known as the standard error of the mean) will be smaller than the population standard deviation and will be equal to the standard deviation of the population divided by the square root of the sample size. If the population has a normal distribution, then the sample means will have a normal distribution. If the population is not normally distributed, but the sample size is sufficiently large, then the sample means will have an approximately normal distribution. Some books define sufficiently large as at least 30 and others as at least 31.

The formula for a z-score when working with the sample means is:

Finite Population Correction Factor If the sample size is more than 5% of the population size and the sampling is done without replacement, then a correction needs to be made to the standard error of the means. In the following, N is the population size and n is the sample size. The adjustment is to multiply the standard error by the square root of the quotient of the difference between the population and sample sizes and one less than the population

size. For the most part, we will be ignoring this in class.

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Stats: Normal Approximation to Binomial

Recall that according to the Central Limit Theorem, the sample mean of any distribution will become approximately normal if the sample size is sufficiently large. It turns out that the binomial distribution can be approximated using the normal distribution if np and nq are both at least 5. Furthermore, recall that the mean of a binomial distribution is np and the variance of the binomial distribution is npq.

Continuity Correction Factor There is a problem with approximating the binomial with the normal. That problem arises because the binomial distribution is a discrete distribution while the normal distribution is a continuous distribution. The basic difference here is that with discrete values, we are talking about heights but no widths, and with the continuous distribution we are talking about both heights and widths. The correction is to either add or subtract 0.5 of a unit from each discrete x-value. This fills in the gaps to make it continuous. This is very similar to expanding of limits to form boundaries that we did with group frequency distributions. Examples Discrete

Continuous

x=6

5.5 < x < 6.5

x>6

x > 6.5

x >= 6

x > 5.5

x<6

x < 5.5

x <= 6

x < 6.5

As you can see, whether or not the equal to is included makes a big difference in the discrete distribution and the way the conversion is performed. However, for a continuous distribution, equality makes no difference. Steps to working a normal approximation to the binomial distribution 1. Identify success, the probability of success, the number of trials, and the desired number of successes. Since this is a binomial problem, these are the same things which were identified when working a binomial problem.

2. Convert the discrete x to a continuous x. Some people would argue that step 3 should be done before this step, but go ahead and convert the x before you forget about it and miss the problem. 3. Find the smaller of np or nq. If the smaller one is at least five, then the larger must also be, so the approximation will be considered good. When you find np, you're actually finding the mean, mu, so denote it as such. 4. Find the standard deviation, sigma = sqrt (npq). It might be easier to find the variance and just stick the square root in the final calculation - that way you don't have to work with all of the decimal places. 5. Compute the z-score using the standard formula for an individual score (not the one for a sample mean). Calculate the probability desi

Stats: Estimation chp8

Definitions Confidence Interval An interval estimate with a specific level of confidence Confidence Level The percent of the time the true mean will lie in the interval estimate given. Consistent Estimator An estimator which gets closer to the value of the parameter as the sample size increases. Degrees of Freedom The number of data values which are allowed to vary once a statistic has been determined. Estimator A sample statistic which is used to estimate a population parameter. It must be unbiased, consistent, and relatively efficient. Interval Estimate A range of values used to estimate a parameter.

Maximum Error of the Estimate The maximum difference between the point estimate and the actual parameter. The Maximum Error of the Estimate is 0.5 the width of the confidence interval for means and proportions. Point Estimate A single value used to estimate a parameter. Relatively Efficient Estimator The estimator for a parameter with the smallest variance. T distribution A distribution used when the population variance is unknown. Unbiased Estimator An estimator whose expected value is the mean of the parameter being estimated. 6. red.

Stats: Introduction to Estimation One area of concern in inferential statistics is the estimation of the population parameter from the sample statistic. It is important to realize the order here. The sample statistic is calculated from the sample data and the population parameter is inferred (or estimated) from this sample statistic. Let me say that again: Statistics are calculated, parameters are estimated. We talked about problems of obtaining the value of the parameter earlier in the course when we talked about sampling techniques. Another area of inferential statistics is sample size determination. That is, how large of a sample should be taken to make an accurate estimation. In these cases, the statistics can't be used since the sample hasn't been taken yet.

Point Estimates

There are two types of estimates we will find: Point Estimates and Interval Estimates. The point estimate is the single best value. A good estimator must satisfy three conditions:   

Unbiased: The expected value of the estimator must be equal to the mean of the parameter Consistent: The value of the estimator approaches the value of the parameter as the sample size increases Relatively Efficient: The estimator has the smallest variance of all estimators which could be used

Confidence Intervals The point estimate is going to be different from the population parameter because due to the sampling error, and there is no way to know who close it is to the actual parameter. For this reason, statisticians like to give an interval estimate which is a range of values used to estimate the parameter. A confidence interval is an interval estimate with a specific level of confidence. A level of confidence is the probability that the interval estimate will contain the parameter. The level of confidence is 1 - alpha. 1-alpha area lies within the confidence interval. Maximum Error of the Estimate

The maximum error of the estimate is denoted by E and is one-half the width of the confidence interval. The basic confidence interval for a symmetric distribution is set up to be the point estimate minus the maximum error of the estimate is less than the true population parameter which is less than the point estimate plus the maximum error of the estimate. This formula will work for means and proportions because they will use the Z or T distributions which are symmetric. Later, we will talk about variances, which don't use a symmetric distribution, and the formula will be different. Area in Tails

Since the level of confidence is 1-alpha, the amount in the tails is alpha. There is a notation in statistics which means the score which has the specified area in the right tail. Examples:

 

Z(0.05) = 1.645 (the Z-score which has 0.05 to the right, and 0.4500 between 0 and it) Z(0.10) = 1.282 (the Z-score which has 0.10 to the right, and 0.4000 between 0 and it).

As a shorthand notation, the () are usually dropped, and the probability written as a subscript. The greek letter alpha is used represent the area in both tails for a confidence interval, and so alpha/2 will be the area in one tail. Here are some common values Confidence Level

Area between 0 and z-score

Area in one tail (alpha/2)

z-score

50%

0.2500

0.2500

0.674

80%

0.4000

0.1000

1.282

90%

0.4500

0.0500

1.645

95%

0.4750

0.0250

1.960

98%

0.4900

0.0100

2.326

99%

0.4950

0.0050

2.576

Notice in the above table, that the area between 0 and the z-score is simply one-half of the confidence level. So, if there is a confidence level which isn't given above, all you need to do to find it is divide the confidence level by two, and then look up the area in the inside part of the Z-table and look up the z-score on the outside. Also notice - if you look at the student's t distribution, the top row is a level of confidence, and the bottom row is the z-score. In fact, this is where I got the extra digit of accuracy from.

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Stats: Estimating the Mean You are estimating the population mean, mu, not the sample mean, x bar.

Population Standard Deviation Known If the population standard deviation, sigma is known, then the mean has a normal (Z) distribution. The maximum error of the estimate is given by the formula for E shown. The Z here is the z-score obtained from the normal table, or the bottom of the t-table as explained in the introduction to estimation. The z-score is a factor of the level of confidence, so you may get in the habit of writing it next to the level of confidence. Once you have computed E, I suggest you save it to the memory on your calculator. On the TI-82, a good choice would be the letter E. The reason for this is that the limits for the confidence interval are now found by subtracting and adding the maximum error of the estimate from/to the sample mean.

Student's t Distribution When the population standard deviation is unknown, the mean has a Student's t distribution. The Student's t distribution was created by William T. Gosset, an Irish brewery worker. The brewery wouldn't allow him to publish his work under his name, so he used the pseudonym "Student". The Student's t distribution is very similar to the standard normal distribution.     

It is symmetric about its mean It has a mean of zero It has a standard deviation and variance greater than 1. There are actually many t distributions, one for each degree of freedom As the sample size increases, the t distribution approaches the normal distribution.

 

It is bell shaped. The t-scores can be negative or positive, but the probabilities are always positive.

Degrees of Freedom

A degree of freedom occurs for every data value which is allowed to vary once a statistic has been fixed. For a single mean, there are n-1 degrees of freedom. This value will change depending on the statistic being used.

Population Standard Deviation Unknown If the population standard deviation, sigma is unknown, then the mean has a student's t (t) distribution and the sample standard deviation is used instead of the population standard deviation. The maximum error of the estimate is given by the formula for E shown. The t here is the t-score obtained from the Student's t table. The t-score is a factor of the level of confidence and the sample size. Once you have computed E, I suggest you save it to the memory on your calculator. On the TI-82, a good choice would be the letter E. The reason for this is that the limits for the confidence interval are now found by subtracting and adding the maximum error of the estimate from/to the sample mean.

Notice the formula is the same as for a population mean when the population standard deviation is known. The only thing that has changed is the formula for the maximum error of the estimate.

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Student's T Critical Values

Conf. Level

50%

80%

90%

95%

98%

99%

One Tail

0.250

0.100

0.050

0.025

0.010

0.005

Two Tail

0.500

0.200

0.100

0.050

0.020

0.010

df = 1

1.000

3.078

6.314

12.706

31.821

63.657

2

0.816

1.886

2.920

4.303

6.965

9.925

3

0.765

1.638

2.353

3.182

4.541

5.841

4

0.741

1.533

2.132

2.776

3.747

4.604

5

0.727

1.476

2.015

2.571

3.365

4.032

6

0.718

1.440

1.943

2.447

3.143

3.707

7

0.711

1.415

1.895

2.365

2.998

3.499

8

0.706

1.397

1.860

2.306

2.896

3.355

9

0.703

1.383

1.833

2.262

2.821

3.250

10

0.700

1.372

1.812

2.228

2.764

3.169

11

0.697

1.363

1.796

2.201

2.718

3.106

12

0.695

1.356

1.782

2.179

2.681

3.055

13

0.694

1.350

1.771

2.160

2.650

3.012

14

0.692

1.345

1.761

2.145

2.624

2.977

15

0.691

1.341

1.753

2.131

2.602

2.947

16

0.690

1.337

1.746

2.120

2.583

2.921

17

0.689

1.333

1.740

2.110

2.567

2.898

18

0.688

1.330

1.734

2.101

2.552

2.878

Conf. Level

50%

80%

90%

95%

98%

99%

One Tail

0.250

0.100

0.050

0.025

0.010

0.005

Two Tail

0.500

0.200

0.100

0.050

0.020

0.010

19

0.688

1.328

1.729

2.093

2.539

2.861

20

0.687

1.325

1.725

2.086

2.528

2.845

21

0.686

1.323

1.721

2.080

2.518

2.831

22

0.686

1.321

1.717

2.074

2.508

2.819

23

0.685

1.319

1.714

2.069

2.500

2.807

24

0.685

1.318

1.711

2.064

2.492

2.797

25

0.684

1.316

1.708

2.060

2.485

2.787

26

0.684

1.315

1.706

2.056

2.479

2.779

27

0.684

1.314

1.703

2.052

2.473

2.771

28

0.683

1.313

1.701

2.048

2.467

2.763

29

0.683

1.311

1.699

2.045

2.462

2.756

30

0.683

1.310

1.697

2.042

2.457

2.750

40

0.681

1.303

1.684

2.021

2.423

2.704

50

0.679

1.299

1.676

2.009

2.403

2.678

60

0.679

1.296

1.671

2.000

2.390

2.660

70

0.678

1.294

1.667

1.994

2.381

2.648

80

0.678

1.292

1.664

1.990

2.374

2.639

90

0.677

1.291

1.662

1.987

2.368

2.632

Conf. Level

50%

80%

90%

95%

98%

99%

One Tail

0.250

0.100

0.050

0.025

0.010

0.005

Two Tail

0.500

0.200

0.100

0.050

0.020

0.010

100

0.677

1.290

1.660

1.984

2.364

2.626

z

0.674

1.282

1.645

1.960

2.326

2.576

Stats: Estimating the Proportion You are estimating the population proportion, p. All estimation done here is based on the fact that the normal can be used to approximate the binomial distribution when np and nq are both at least 5. Thus, the p that were talking about is the probability of success on a single trial from the binomial experiments.

Recall:

The best point estimate for p is p hat, the sample proportion: If the formula for z is divided by n in both the numerator and the denominator, then

the formula for z becomes:

Solving this for p to come up with a confidence interval, gives the maximum error of

the estimate as:

.

This is not, however, the formula that we will use. The problem with estimation is that you don't know the value of the parameter (in this case p), so you can't use it to estimate itself - if you knew it, then there would be no problem to work out. So we will replace the parameter by the statistic in the formula for the maximum error of the estimate. The maximum error of the estimate is given by the formula for E shown. The Z here is the z-score obtained from the normal table, or the bottom of the t-table as explained in the introduction to estimation. The z-score is a factor of the level of confidence, so you may get in the habit of writing it next to the level of confidence. When you're computing E, I suggest that you find the sample proportion, p hat, and save it to P on the calculator. This way, you can find q as (1-p). Do NOT round the value for p hat and use the rounded value in the calculations. This will lead to error. Once you have computed E, I suggest you save it to the memory on your calculator. On the TI-82, a good choice would be the letter E. The reason for this is that the limits for the confidence interval are now found by subtracting and adding the maximum error of the estimate from/to the sample proportion.

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Stats: Sample Size Determination chp9 The sample size determination formulas come from the formulas for the maximum error of the estimates. The formula is solved for n. Be sure to round the answer obtained up to the next whole number, not off to the nearest whole number. If you round off, then you will exceed your maximum error of the estimate in some cases.

By rounding up, you will have a smaller maximum error of the estimate than allowed, but this is better than having a larger one than desired.

Population Mean Here is the formula for the sample size which is obtained by solving the maximum error of the estimate formula for the population mean for n.

Population Proportion Here is the formula for the sample size which is obtained by solving the maximum error of the estimate formula for the population proportion for n. Some texts use p hat and q hat, but since the sample hasn't been taken, there is no value for the sample proportion. p and q are taken from a previous study, if one is available. If there is no previous study or estimate available, then use 0.5 for p and q, as these are the values which will give the largest sample size, and it is better to have too large of a sample size and come under the maximum error of the estimate than to have too small of a sample size and exceed the maximum error of the estimate.

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Stats: Hypothesis Testing Definitions Null Hypothesis ( H0 ) Statement of zero or no change. If the original claim includes equality (<=, =, or >=), it is the null hypothesis. If the original claim does not include equality (<, not

then the null hypothesis is the complement of the original claim. The null hypothesis always includes the equal sign. The decision is based on the null hypothesis. equal, >)

Alternative Hypothesis ( H1 or Ha ) Statement which is true if the null hypothesis is false. The type of test (left, right, or two-tail) is based on the alternative hypothesis. Type I error Rejecting the null hypothesis when it is true (saying false when true). Usually the more serious error. Type II error Failing to reject the null hypothesis when it is false (saying true when false). alpha Probability of committing a Type I error. beta Probability of committing a Type II error. Test statistic Sample statistic used to decide whether to reject or fail to reject the null hypothesis. Critical region Set of all values which would cause us to reject H0 Critical value(s) The value(s) which separate the critical region from the non-critical region. The critical values are determined independently of the sample statistics. Significance level ( alpha ) The probability of rejecting the null hypothesis when it is true. alpha = 0.05 and alpha = 0.01 are common. If no level of significance is given, use alpha = 0.05. The level of significance is the complement of the level of confidence in estimation.

Decision A statement based upon the null hypothesis. It is either "reject the null hypothesis" or "fail to reject the null hypothesis". We will never accept the null hypothesis. Conclusion A statement which indicates the level of evidence (sufficient or insufficient), at what level of significance, and whether the original claim is rejected (null) or ed (alternative).

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Stats: Hypothesis Testing Introduction Be sure to read through the definitions for this section before trying to make sense out of the following. The first thing to do when given a claim is to write the claim mathematically (if possible), and decide whether the given claim is the null or alternative hypothesis. If the given claim contains equality, or a statement of no change from the given or accepted condition, then it is the null hypothesis, otherwise, if it represents change, it is the alternative hypothesis. The following example is not a mathematical example, but may help introduce the concept.

Example "He's dead, Jim," said Dr. McCoy to Captain Kirk.

Mr. Spock, as the science officer, is put in charge of statistically determining the correctness of Bones' statement and deciding the fate of the crew member (to vaporize or try to revive) His first step is to arrive at the hypothesis to be tested. Does the statement represent a change in previous condition?  

Yes, there is change, thus it is the alternative hypothesis, H1 No, there is no change, therefore is the null hypothesis, H0

The correct answer is that there is change. Dead represents a change from the accepted state of alive. The null hypothesis always represents no change. Therefore, the hypotheses are:  

H0 : Patient is alive. H1 : Patient is not alive (dead).

States of nature are something that you, as a statistician have no control over. Either it is, or it isn't. This represents the true nature of things. Possible states of nature (Based on H0)  

Patient is alive (H0 true - H1 false ) Patient is dead (H0 false - H1 true)

Decisions are something that you have control over. You may make a correct decision or an incorrect decision. It depends on the state of nature as to whether your decision is correct or in error. Possible decisions (Based on H0 ) / conclusions (Based on claim )  

Reject H0 / "Sufficient evidence to say patient is dead" Fail to Reject H0 / "Insufficient evidence to say patient is dead"

There are four possibilities that can occur based on the two possible states of nature and the two decisions which we can make. Statisticians will never accept the null hypothesis, we will fail to reject. In other words, we'll say that it isn't, or that we don't have enough evidence to say that it isn't, but we'll never say that it is, because someone else might come along with another sample which shows that it isn't and we don't want to be wrong.

Statistically (double) speaking ... State of Nature Decision

H0 True

H0 False

Reject H0

Patient is alive,

Patient is dead, Sufficient evidence of death

Sufficient evidence of death

Fail to reject H0

Patient is alive,

Patient is dead, Insufficient evidence of death

Insufficient evidence of death

In English ... State of Nature Decision

H0 True

H0 False

Reject H0

Vaporize a live person

Vaporize a dead person

Fail to reject H0

Try to revive a live person

Try to revive a dead person

Were you right ? ... State of Nature

Decision

H0 True

H0 False

Reject H0

Type I Error alpha

Correct Assessment

Fail to reject H0

Correct Assessment

Type II Error beta

Which of the two errors is more serious? Type I or Type II ? Since Type I is the more serious error (usually), that is the one we concentrate on. We usually pick alpha to be very small (0.05, 0.01). Note: alpha is not a Type I error. Alpha is the probability of committing a Type I error. Likewise beta is the probability of committing a Type II error. Conclusions

Conclusions are sentence answers which include whether there is enough evidence or not (based on the decision), the level of significance, and whether the original claim is ed or rejected. Conclusions are based on the original claim, which may be the null or alternative hypotheses. The decisions are always based on the null hypothesis Original Claim

Decision Reject H0 "SUFFICIENT"

H0 "REJECT"

H1 ""

There is sufficient evidence at the alpha level of significance to reject the claim that (insert original claim here)

There is sufficient evidence at the alpha level of significance to the claim that (insert original claim here)

Fail to reject H0 "INSUFFICIENT"

There is insufficient evidence at the alpha level of significance to rejectthe claim that (insert original claim here)

There is insufficient evidence at the alpha level of significance to the claim that (insert original claim here)

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Stats: Type of Tests This document will explain how to determine if the test is a left tail, right tail, or twotail test. The type of test is determined by the Alternative Hypothesis ( H1 )

Left Tailed Test H1: parameter < value Notice the inequality points to the left

Decision Rule: Reject H0 if t.s. < c.v.

Right Tailed Test H1: parameter > value Notice the inequality points to the right

Decision Rule: Reject H0 if t.s. > c.v.

Two Tailed Test H1: parameter not equal value Another way to write not equal is < or > Notice the inequality points to both sides

Decision Rule: Reject H0 if t.s. < c.v. (left) or t.s. > c.v. (right)

The decision rule can be summarized as follows: Reject H0 if the test statistic falls in the critical region (Reject H0 if the test statistic is more extreme than the critical value)

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Stats: Confidence Intervals as Tests Using the confidence interval to perform a hypothesis test only works with a twotailed test. 



If the hypothesized value of the parameter lies within the confidence interval with a 1-alpha level of confidence, then the decision at an alpha level of significance is to fail to reject the null hypothesis. If the hypothesized value of the parameter lies outside the confidence interval with a 1-alpha level of confidence, then the decision at an alpha level of significance is to reject the null hypothesis.

Sounds simple enough, right? It is. However, it has a couple of problems.  

It only works with two-tail hypothesis tests. It requires that you compute the confidence interval first. This involves taking a z-score or t-score and converting it into an x-score, which is more difficult than standardizing an x-score.

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Stats: Hypothesis Testing Steps

Here are the steps to performing hypothesis testing 1. Write the original claim and identify whether it is the null hypothesis or the alternative hypothesis. 2. Write the null and alternative hypothesis. Use the alternative hypothesis to identify the type of test. 3. Write down all information from the problem. 4. Find the critical value using the tables 5. Compute the test statistic 6. Make a decision to reject or fail to reject the null hypothesis. A picture showing the critical value and test statistic may be useful. 7. Write the conclusion.

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Stats: Testing a Single Proportion

You are testing p, you are not testing p hat. If you knew the value of p, then there would be nothing to test.

All hypothesis testing is done under the assumption the null hypothesis is true! I can't emphasize this enough. The value for all population parameters in the test statistics come from the null hypothesis. This is true not only for proportions, but all of the testing we're going to be doing. The population proportion has an approximately normal distribution if np and nq are both at least 5. that we are approximating the binomial using the normal, and that the p we're talking about is the probability of success on a single trial. The test statistic is shown in the box to the right. The critical value is found from the normal table, or from the bottom row of the ttable. The steps involved in the hypothesis testing remain the same. The only thing that changes is the formula for calculating the test statistic and perhaps the distribution which is used.

General Pattern Notice the general pattern of these test statistics is (observed - expected) / standard deviation.

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Stats: Testing a Single Proportion

You are testing p, you are not testing p hat. If you knew the value of p, then there would be nothing to test.

All hypothesis testing is done under the assumption the null hypothesis is true! I can't emphasize this enough. The value for all population parameters in the test statistics come from the null hypothesis. This is true not only for proportions, but all of the testing we're going to be doing. The population proportion has an approximately normal distribution if np and nq are both at least 5. that we are approximating the binomial using the normal, and that the p we're talking about is the probability of success on a single trial. The test statistic is shown in the box to the right. The critical value is found from the normal table, or from the bottom row of the ttable. The steps involved in the hypothesis testing remain the same. The only thing that changes is the formula for calculating the test statistic and perhaps the distribution which is used.

General Pattern Notice the general pattern of these test statistics is (observed - expected) / standard deviation.

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Stats: Probability Values

Classical Approach The Classical Approach to hypothesis testing is to compare a test statistic and a critical value. It is best used for distributions which give areas and require you to look up the critical value (like the Student's t distribution) rather than distributions which have you look up a test statistic to find an area (like the normal distribution). The Classical Approach also has three different decision rules, depending on whether it is a left tail, right tail, or two tail test. One problem with the Classical Approach is that if a different level of significance is desired, a different critical value must be read from the table.

P-Value Approach The P-Value Approach, short for Probability Value, approaches hypothesis testing from a different manner. Instead of comparing z-scores or t-scores as in the classical approach, you're comparing probabilities, or areas. The level of significance (alpha) is the area in the critical region. That is, the area in the tails to the right or left of the critical values. The p-value is the area to the right or left of the test statistic. If it is a two tail test, then look up the probability in one tail and double it. If the test statistic is in the critical region, then the p-value will be less than the level of significance. It does not matter whether it is a left tail, right tail, or two tail test. This rule always holds. Reject the null hypothesis if the p-value is less than the level of significance. You will fail to reject the null hypothesis if the p-value is greater than or equal to the level of significance. The p-value approach is best suited for the normal distribution when doing calculations by hand. However, many statistical packages will give the p-value but not the critical value. This is because it is easier for a computer or calculator to find the probability than it is to find the critical value.

Another benefit of the p-value is that the statistician immediately knows at what level the testing becomes significant. That is, a p-value of 0.06 would be rejected at an 0.10 level of significance, but it would fail to reject at an 0.05 level of significance. Warning: Do not decide on the level of significance after calculating the test statistic and finding the p-value. Here is a proportion to help you keep the order straight. Any proportion equivalent to the following statement is correct. The test statistic is to the p-value as the critical value is to the level of significance.

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Stats: Two Parameter Testing chp10 Definitions Dependent Samples Samples in which the subjects are paired or matched in some way. Dependent samples must have the same sample size, but it is possible to have the same sample size without being dependent. Independent Samples Samples which are independent when they are not related. Independent samples may or may not have the same sample size. Pooled Estimate of the Variance A weighted average of the two sample variances when the variances are equal. The variances are "close enough" to be considered equal, but not exactly the same, so this pooled estimate brings the two together to find the average variance.

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Stats: Dependent Means

There are two possible cases when testing two population means, the dependent case and the independent case. Most books treat the independent case first, but I'm putting the dependent case first because it follows immediately from the test for a single population mean in the previous chapter.

The Mean of the Difference: The idea with the dependent case is to create a new variable, D, which is the difference between the paired values. You will then be testing the mean of this new variable. Here are some steps to help you accomplish the hypothesis testing 1. Write down the original claim in simple . For example: After > Before. 2. Move everything to one side: After - Before > 0. 3. Call the difference you have on the left side D: D = After - Before > 0. 4. Convert to proper notation: 5. Compute the new variable D and be sure to follow the order you have defined in step 3. Do not simply take the smaller away from the larger. From this point, you can think of having a new set of values. Technically, they are called D, but you can think of them as x. The original values from the two samples can be discarded. 6. Find the mean and standard deviation of the variable D. Use these as the values in the t-test from chapter 9.

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Stats: Independent Means

Sums and Differences of Independent Variables Independent variables can be combined to form new variables. The mean and variance of the combination can be found from the means and the variances of the original variables. Combination of Variables

In English (Melodic Mathematics)

The mean of a sum is the sum of the means. The mean of a difference is the difference of the means. The variance of a sum is the sum of the variances. The variance of a difference is the sum of the variances.

The Difference of the Means: Since we are combining two variables by subtraction, the important rules from the table above are that the mean of the difference is the difference of the means and the variance of the difference is the sum of the variances. It is important to note that the variance of the difference is the sum of the variances, not the standard deviation of the difference is the sum of the standard deviations.

When we go to find the standard error, we must combine variances to do so. Also, you're probably wondering why the variance of the difference is the sum of the variances instead of the difference of the variances. Since the values are squared, the negative associated with the second variable becomes positive, and it becomes the sum of the variances. Also, variances can't be negative, and if you took the difference of the variances, it could be negative. Population Variances Known

When the population variances are known, the difference of the means has a normal distribution. The variance of the difference is the sum of the variances divided by the sample sizes. This makes sense, hopefully, because according to the central limit theorem, the variance of the sampling distribution of the sample means is the variance divided by the sample size, so what we are doing is add the variance of each mean together. The test statistic is shown. Population Variances Unknown, but both sample sizes large

When the population variances aren't known, the difference of the means has a Student's t distribution. However, if both sample sizes are large enough, then you will be using the normal row from the t-table, so your book lumps this under the normal distribution, rather than the t-distribution. This gives us the chance to work the problem without knowing if the population variances are equal or not. The test statistic is shown, and is identical to above, except the sample variances are used instead of the population variances. Population Variances Unknown, unequal with small sample sizes

Ok, you're probably wondering how do you know if the variances are equal or not if you don't know what they are. Some books teach the F-test to test the equality of two variances, and if your book does that, then you should use the F-test to see. Other books (statisticians) argue that if you do the F-test first to see if the variances are equal, and then use the same level of significance to perform the t-test to test the difference of the means, that the overall level of significance isn't the same. So, the Bluman text tells the student whether or not the variances are equal and the Triola text.

Since you don't know the population variances, you're going to be using a Student's t distribution. Since the variances are unequal, there is no attempt made to average them together as we will in the next situation. The degrees of freedom is the smaller of the two degrees of freedom (n-1 for each). The "min" function means take the minimum or smaller of the two values. Otherwise, the formula is the same as we used with large sample sizes. Population Variances Unknown but equal with small sample sizes

If the variances are equal, then an effort is made to average them together. Now, equal does not mean identical. It is possible for two variances to be statistically equal but be numerically different. We will find a pooled estimate of the variance which is simply the weighted mean of the variance. The weighting factors are the degrees of freedom. Once the pooled estimate of the variance is computed, this mean (average) variance is used in the place of the individual sample variances. Otherwise, the formula is the same as before. The degrees of freedom are the sum of the individual degrees of freedom.

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Stats: Two Proportions that the normal distribution can be used to approximate the binomial distribution in certain cases. Specifically, the approximation was considered good when np and nq were both at least 5. Well, now, we're talking about two proportions, so np and nq must be at least 5 for both samples.

We don't have a way to specifically test two proportions for values, what we have is the ability to test the difference between the proportions. So, much like the test for two means from independent populations, we will be looking at the difference of the proportions. We will also be computing an average proportion and calling it pbar. It is the total number of successes divided by the total number of trials. The definitions which are necessary are shown to the right. The test statistic has the same general pattern as before (observed minus expected divided by standard error). The test statistic used here is similar to that for a single population proportion, except the difference of proportions are used instead of a single proportion, and the value of p-bar is used instead of p in the standard error portion. Since we're using the normal approximation to the binomial, the difference of proportions has a normal distribution. The test statistic is given. Some people will be tempted to try to simplify the denominator of this test statistic incorrectly. It can be simplified, but the correct simplification is not to simply place the product of p-bar and q-bar over the sum of the n's. that to add fractions, you must have a common denominator, that is why this simplification is incorrect. The correct simplification would be to factor a p-bar and q-bar out of the two expressions. This is usually the formula given, because it is easier to calculate, but I wanted to give it the other way first so you could compare it to the other formulas and see how similar they all are.

Stats: Correlation & Regression chp11 Definitions Coefficient of Determination The percent of the variation that can be explained by the regression equation

Correlation A method used to determine if a relationship between variables exists Correlation Coefficient A statistic or parameter which measures the strength and direction of a relationship between two variables Dependent Variable A variable in correlation or regression that can not be controlled, that is, it depends on the independent variable. Independent Variable A variable in correlation or regression which can be controlled, that is, it is independent of the other variable. Pearson Product Moment Correlation Coefficient A measure of the strength and direction of the linear relationship between two variables Regression A method used to describe the relationship between two variables. Regression Line The best fit line. Scatter Plot An plot of the data values on a coordinate system. The independent variable is graphed along the x-axis and the dependent variable along the y-axis Standard Error of the Estimate The standard deviation of the observed values about the predicted values

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TI-82: Scatter Plots, Regression Lines You can use the calculator to draw scatter plots. See the instructions on using the calculator to do statistics and lists. This provides an overview as well as some helpful advice for working with statistics on the calculator.

Scatter Plots 1. 2. 3. 4. 5. 6. 7. 8.

Enter the x values into L1 and the y variables into L2. Go to Stat Plot (2nd y=) Turn Plot 1 on Choose the type to be scatter plot (1st type) Set Xlist to L1 Set Ylist to L2 Set the Mark to any of the three choices Zoom to the Stat setting (#9)

Note, the Ylist and Mark won't show up until you select a scatter plot

Regression Lines 1. 2. 3. 4. 5. 6. 7.

Setup the scatter plot as instructed above Go into the Stats, Calc, Setup screen Setup the 2-Var Stats so that: Xlist = L1, Ylist = L2, Freq = 1 Calculate the Linear Regression (ax+b) (#5) Go into the Plot screen. Position the cursor on the Y1 plot and hit CLEAR to erase it. While still in the Y1 data entry field, go to the VARS, STATS, EQ screen and choose option 7 which is the regression equation 8. Hit GRAPH

Regression Lines, part 2 The above technique works, but it requires that you change the equation being graphed every time you change problems. It is possible to stick the regression equation "ax+b" into the Y1 plot and then it will automatically graph the correct regression equation each time.

Do this once

1. 2. 3. 4.

Setup the scatter plot as instructed above Go into the Plot screen. Position the cursor on the Y1 plot and hit CLEAR to erase it. Enter a*x+b into the function. The a and b can be found under the VARS, STATS, EQ screen

Do this for each graph

1. 2. 3. 4.

Go into the Stats, Calc, Setup screen Setup the 2-Var Stats so that: Xlist = L1, Ylist = L2, Freq = 1 Calculate the Linear Regression (ax+b) (#5) Hit the GRAPH key

It is important that you calculate the linear regression variables before trying to graph the regression line. If you change the data in the lists or have not calculated the linear regression equations, then you will get an " ERR: Undefined" when you try to graph the data. Be sure to turn off the stats plots and/or the Y1 plot when you need to graph other data.

Stats: Correlation Sum of Squares We introduced a notation earlier in the course called the sum of squares. This notation was the SS notation, and will make these formulas much easier to work with.

Notice these are all the same pattern,

SS(x) could be written as

Also note that

Pearson's Correlation Coefficient There is a measure of linear correlation. The population parameter is denoted by the greek letter rho and the sample statistic is denoted by the roman letter r. Here are some properties of r     



r only measures the strength of a linear relationship. There are other kinds of relationships besides linear. r is always between -1 and 1 inclusive. -1 means perfect negative linear correlation and +1 means perfect positive linear correlation r has the same sign as the slope of the regression (best fit) line r does not change if the independent (x) and dependent (y) variables are interchanged r does not change if the scale on either variable is changed. You may multiply, divide, add, or subtract a value to/from all the x-values or y-values without changing the value of r. r has a Student's t distribution

Here is the formula for r. Don't worry about it, we won't be finding it this way. This formula can be simplified through some simple algebra and then some substitutions using the SS notation discussed earlier.

If you divide the numerator and denominator by n, then you get something which is starting to hopefully look familiar. Each of these values have been seen before in the Sum of Squares notation section. So, the linear correlation coefficient can be written in of sum of squares. This is the formula that we would be using for calculating the linear correlation coefficient if we were doing it by hand. Luckily for us, the TI-82 has this calculation built into it, and we won't have to do it by hand at all.

Hypothesis Testing The claim we will be testing is "There is significant linear correlation" The Greek letter for r is rho, so the parameter used for linear correlation is rho  

H0: rho = 0 H1: rho <> 0

r has a t distribution with n-2 degrees of freedom, and the test statistic is given by: Now, there are n-2 degrees of freedom this time. This is a difference from before. As an over-simplification, you subtract one degree of freedom for each variable, and since there are 2 variables, the degrees of freedom are n-2.

This doesn't look like our

If you consider the standard error for r is

the formula for the test statistic is looking for.

, which does look like the pattern we're

that Hypothesis testing is always done under the assumption that the null hypothesis is true.

Since H0 is rho = 0, this formula is equivalent to the one given in the book. Additional Note: 1-r2 is later identified as the coefficient of non-determination

Hypothesis Testing Revisited If you are testing to see if there is significant linear correlation (a two tailed test), then there is another way to perform the hypothesis testing. There is a table of critical values for the Pearson's Product Moment Coefficient (PPMC) given in the text book. The degrees of freedom are n-2. The test statistic in this case is simply the value of r. You compare the absolute value of r (don't worry if it's negative or positive) to the critical value in the table. If the test statistic is greater than the critical value, then there is significant linear correlation. Furthermore, you are able to say there is significant positive linear correlation if the original value of r is positive, and significant negative linear correlation if the original value of r was negative. This is the most common technique used. However, the first technique, with the tvalue must be used if it is not a two-tail test, or if a different level of significance (other than 0.01 or 0.05) is desired.

Causation If there is a significant linear correlation between two variables, then one of five situations can be true.   

There is a direct cause and effect relationship There is a reverse cause and effect relationship The relationship may be caused by a third variable

 

The relationship may be caused by complex interactions of several variables The relationship may be coincidental

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Stats: Regression The idea behind regression is that when there is significant linear correlation, you can use a line to estimate the value of the dependent variable for certain values of the independent variable. The regression equation should only used  





When there is significant linear correlation. That is, when you reject the null hypothesis that rho=0 in a correlation hypothesis test. The value of the independent variable being used in the estimation is close to the original values. That is, you should not use a regression equation obtained using x's between 10 and 20 to estimate y when x is 200. The regression equation should not be used with different populations. That is, if x is the height of a male, and y is the weight of a male, then you shouldn't use the regression equation to estimate the weight of a female. The regression equation shouldn't be used to forecast values not from that time frame. If data is from the 1960's, it probably isn't valid in the 1990's.

Assuming that you've decided that you can have a regression equation because there is significant linear correlation between the two variables, the equation becomes: y' = ax + b or y' = a + bx (some books use y-hat instead of y-prime). The Bluman text uses the second formula, however, more people are familiar with the notion of y = mx + b, so I will use the first.

a is the slope of the regression line:

b is the y-intercept of the regression line:

The regression line is sometimes called the "line of best fit" or the "best fit line". Since it "best fits" the data, it makes sense that the line es through the means. The regression equation is the line with slope a ing through the point Another way to write the equation would be

apply just a little algebra, and we have the formulas for a and b that we would use (if we were stranded on a desert island without the TI-82) ...

It also turns out that the slope of the regression line can be written as . Since the standard deviations can't be negative, the sign of the slope is determined by the sign of the correlation coefficient. This agrees with the statement made earlier that the slope of the regression line will have the same slope as the correlation coefficient.

TI-82 Luckily, the TI-82 will find these values for us (isn't it a wonderful calculator?). We can also use the TI-82 to plot the regression line on the scatter plot.

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TI-82: Correlation / Regression See the instructions on using the calculator to do statistics and lists. This provides an overview as well as some helpful advice for working with statistics on the calculator.

Calculating Values

1. 2. 3. 4.

Enter the data. Put the x-values into list 1 and the y-values into list 2. Go into the Stats, Calc, Setup screen Setup the 2-Var Stats so that: Xlist = L1, Ylist = L2, Freq = 1 Calculate the Linear Regression (ax+b) (#5)

This screen will give you the sample linear correlation coefficient, r; the slope of the regression equation, a; and the y-intercept of the regression equation, b. Just record the value of r. To write the regression equation, replace the values of a and b into the equation "y-hat = ax+b". To find the coefficient of determination, square r. You can find the variable r under VARS, STATS, EQ, r (#6).

Stats: Coefficient of Determination

Coefficient of Determination The coefficient of determination is ...   

the percent of the variation that can be explained by the regression equation. the explained variation divided by the total variation the square of r

What's all this variation stuff? Every sample has some variation in it (unless all the values are identical, and that's unlikely to happen). The total variation is made up of two parts, the part that can be explained by the regression equation and the part that can't be explained by the regression equation.

Well, the ratio of the explained variation to the total variation is a measure of how good the regression line is. If the regression line ed through every point on the scatter plot exactly, it would be able to explain all of the variation. The further the line is from the points, the less it is able to explain.

Coefficient of Non-Determination The coefficient of non-determination is ...   

The percent of variation which is unexplained by the regression equation The unexplained variation divided by the total variation 1 - r^2

Standard Error of the Estimate The coefficient of non-determination was used in the t-test to see if there was significant linear correlation. It was the in the numerator of the standard error formula. The standard error of the estimate is the square root of the coefficient of non-determination divided by it's degrees of freedom.

Confidence Interval for y' The following only works when the sample size is large. Large in this instance is usually taken to be more than 100. We're not going to cover this in class, but is provided here for your information. The maximum error of the estimate is given, and this maximum error of the estimate is subtracted from and added to the estimated value of y.

Stats: Chi-Square DefinitionsStats:

Chi-Square chp12

Definitions Chi-square distribution A distribution obtained from the multiplying the ratio of sample variance to population variance by the degrees of freedom when random samples are selected from a normally distributed population Contingency Table Data arranged in table form for the chi-square independence test Expected Frequency The frequencies obtained by calculation. Goodness-of-fit Test A test to see if a sample comes from a population with the given distribution. Independence Test A test to see if the row and column variables are independent. Observed Frequency The frequencies obtained by observation. These are the sample frequencies.

Table of Contents Chi-square distribution A distribution obtained from the multiplying the ratio of sample variance to population variance by the degrees of freedom when random samples are selected from a normally distributed population

Contingency Table Data arranged in table form for the chi-square independence test Expected Frequency The frequencies obtained by calculation. Goodness-of-fit Test A test to see if a sample comes from a population with the given distribution. Independence Test A test to see if the row and column variables are independent. Observed Frequency The frequencies obtained by observation. These are the sample frequencies.

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Stats: Chi-Square Distribution The chi-square ( ) distribution is obtained from the values of the ratio of the sample variance and population variance multiplied by the degrees of freedom. This occurs when the population is normally distributed with population variance sigma^2.

Properties of the Chi-Square  

Chi-square is non-negative. Is the ratio of two non-negative values, therefore must be non-negative itself. Chi-square is non-symmetric.

 

There are many different chi-square distributions, one for each degree of freedom. The degrees of freedom when working with a single population variance is n-1.

Chi-Square Probabilities Since the chi-square distribution isn't symmetric, the method for looking up left-tail values is different from the method for looking up right tail values.   

Area to the right - just use the area given. Area to the left - the table requires the area to the right, so subtract the given area from one and look this area up in the table. Area in both tails - divide the area by two. Look up this area for the right critical value and one minus this area for the left critical value.

DF which aren't in the table When the degrees of freedom aren't listed in the table, there are a couple of choices that you have. 



You can interpolate. This is probably the more accurate way. Interpolation involves estimating the critical value by figuring how far the given degrees of freedom are between the two df in the table and going that far between the critical values in the table. Most people born in the 70's didn't have to learn interpolation in high school because they had calculators which would do logarithms (we had to use tables in the "good old" days). You can go with the critical value which is less likely to cause you to reject in error (type I error). For a right tail test, this is the critical value further to the right (larger). For a left tail test, it is the value further to the left (smaller). For a two-tail test, it's the value further to the left and the value further to the right. Note, it is not the column with the degrees of freedom further to the right, it's the critical value which is further to the right. The Bluman text has this wrong on page 422. The guideline is right, the instructions are wrong.

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Stats: Single Population Variance

The variable has a chi-square distribution if the population variance has a normal distribution. The degrees of freedom are n-1. We can use this to test the population variance under certain conditions

Conditions for testing    

The population has a normal distribution The data is from a random sample The observations must be independent of each other The test statistic has a chi-square distribution with n-1 degrees of freedom and is given by:

Testing is done in the same manner as before. , all hypothesis testing is done under the assumption the null hypothesis is true.

Confidence Intervals If you solve the test statistic formula for the population variance, you get: 1. Find the two critical values (alpha/2 and 1-alpha/2) 2. Compute the value for the population variance given above. 3. Place the population variance between the two values calculated in step 2 (put the smaller one first). Note, the left-hand endpoint of the confidence interval comes when the right critical value is used and the right-hand endpoint of the confidence interval comes when the left critical value is used. This is because the critical values are in the denominator and so dividing by the larger critical value (right tail) gives the smaller endpoint.

Stats: Goodness-of-fit Test

The idea behind the chi-square goodness-of-fit test is to see if the sample comes from the population with the claimed distribution. Another way of looking at that is to ask if the frequency distribution fits a specific pattern. Two values are involved, an observed value, which is the frequency of a category from a sample, and the expected frequency, which is calculated based upon the claimed distribution. The derivation of the formula is very similar to that of the variance which was done earlier (chapter 2 or 3). The idea is that if the observed frequency is really close to the claimed (expected) frequency, then the square of the deviations will be small. The square of the deviation is divided by the expected frequency to weight frequencies. A difference of 10 may be very significant if 12 was the expected frequency, but a difference of 10 isn't very significant at all if the expected frequency was 1200. If the sum of these weighted squared deviations is small, the observed frequencies are close to the expected frequencies and there would be no reason to reject the claim that it came from that distribution. Only when the sum is large is the a reason to question the distribution. Therefore, the chi-square goodness-of-fit test is always a right tail test. The test statistic has a chi-square distribution when the following assumptions are met  

The data are obtained from a random sample The expected frequency of each category must be at least 5. This goes back to the requirement that the data be normally distributed. You're simulating a multinomial experiment (using a discrete distribution) with the goodness-of-fit test (and a continuous distribution), and if each expected frequency is at least five then you can use the normal distribution to approximate (much like the binomial). If the expected

The following are properties of the goodness-of-fit test     

The data are the observed frequencies. This means that there is only one data value for each category. Therefore, ... The degrees of freedom is one less than the number of categories, not one less than the sample size. It is always a right tail test. It has a chi-square distribution. The value of the test statistic doesn't change if the order of the categories is switched.



The test statistic is

Interpreting the Claim There are four ways you might be given a claim. 1. The values occur with equal frequency. Other words for this are "uniform", "no preference", or "no difference". To find the expected frequencies, total the observed frequencies and divide by the number of categories. This quotient is the expected frequency for each category. 2. Specific proportions or probabilities are given. To find the expected frequencies, multiply the total of the observed frequencies by the probability for each category. 3. The expected frequencies are given to you. In this case, you don't have to do anything. 4. A specific distribution is claimed. For example, "The data is normally distributed". To work a problem like this, you need to group the data and find the frequency for each class. Then, find the probability of being within that class by converting the scores to z-scores and looking up the probabilities. Finally, multiply the probabilities by the total observed frequency. (It's not really as bad as it sounds).

Using the TI-82 You can use the lists on the TI-82 to perform the chi-square goodness-of-fit test.

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TI-82: Goodness-of-Fit You can perform a chi-square goodness-of-fit test using the TI-82. Here are the steps. 1. Enter the observed frequencies into List 1. 2. Enter the expected frequencies into List 2. a. If you're given the expected frequencies, enter them into List 2.

b. If you're given probabilities, then enter the probabilities into List 2 and multiply List 2 by the sum of List 1 and replace List 2 with that product: sum L1 * L2 -> L2 c. If you're testing that all categories appear with equal frequency, then you can a) enter that value into List 2, or b) enter the total frequency into each value of List 2 and then divide the list by the number of categories: L2 / k -> L2 (replace k by the number of categories), or c) enter 1 for each value in List 2 and then multiply the list by the common expected frequency: L2 * E -> L2 (replace E by the expected frequency) 3. Calculate the test statistic: sum ((L1 - L2)^2 / L2)

Stats: Test for Independence In the test for independence, the claim is that the row and column variables are independent of each other. This is the null hypothesis. The multiplication rule said that if two events were independent, then the probability of both occurring was the product of the probabilities of each occurring. This is key to working the test for independence. If you end up rejecting the null hypothesis, then the assumption must have been wrong and the row and column variable are dependent. , all hypothesis testing is done under the assumption the null hypothesis is true. The test statistic used is the same as the chi-square goodness-of-fit test. The principle behind the test for independence is the same as the principle behind the goodness-offit test. The test for independence is always a right tail test. In fact, you can think of the test for independence as a goodness-of-fit test where the data is arranged into table form. This table is called a contingency table. The test statistic has a chi-square distribution when the following assumptions are met  

The data are obtained from a random sample The expected frequency of each category must be at least 5.

The following are properties of the test for independence 

The data are the observed frequencies.

 

    



The data is arranged into a contingency table. The degrees of freedom are the degrees of freedom for the row variable times the degrees of freedom for the column variable. It is not one less than the sample size, it is the product of the two degrees of freedom. It is always a right tail test. It has a chi-square distribution. The expected value is computed by taking the row total times the column total and dividing by the grand total The value of the test statistic doesn't change if the order of the rows or columns are switched. The value of the test statistic doesn't change if the rows and columns are interchanged (transpose of the matrix) The test statistic is

Using the TI-82 There is a program called CONTING (for contingency table) for the TI-82 which will compute the test statistic for you. You still need to look up the critical value in the table.

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CONTING Program This program completes a test for independence using a contingency table. The observed frequencies must be contained in matrix [A]and the result is a test statistic having a chi-square distribution. When the program is done running, the following variables are defined    

List 1 contains the observed frequencies List 2 contains the expected frequencies List 3 contains the row totals List 4 contains the column totals

Stats: F-Test chp13 Definitions F-distribution The ratio of two independent chi-square variables divided by their respective degrees of freedom. If the population variances are equal, this simplifies to be the ratio of the sample variances. Analysis of Variance (ANOVA) A technique used to test a hypothesis concerning the means of three or mor populations. One-Way Analysis of Variance Analysis of Variance when there is only one independent variable. The null hypothesis will be that all population means are equal, the alternative hypothesis is that at least one mean is different. Between Group Variation The variation due to the interaction between the samples, denoted SS(B) for Sum of Squares Between groups. If the sample means are close to each other (and therefore the Grand Mean) this will be small. There are k samples involved with one data value for each sample (the sample mean), so there are k-1 degrees of freedom. Between Group Variance The variance due to the interaction between the samples, denoted MS(B) for Mean Square Between groups. This is the between group variation divided by its degrees of freedom. Within Group Variation The variation due to differences within individual samples, denoted SS(W) for Sum of Squares Within groups. Each sample is considered independently, no interaction between samples is involved. The degrees of freedom is equal to

the sum of the individual degrees of freedom for each sample. Since each sample has degrees of freedom equal to one less than their sample sizes, and there are k samples, the total degrees of freedom is k less than the total sample size: df = N - k. Within Group Variance The variance due to the differences within individual samples, denoted MS(W) for Mean Square Within groups. This is the within group variation divided by its degrees of freedom. Scheffe' Test A test used to find where the differences between means lie when the Analysis of Variance indicates the means are not all equal. The Scheffe' test is generally used when the sample sizes are different. Tukey Test A test used to find where the differences between the means lie when the Analysis of Variance indicates the means are not all equal. The Tukey test is generally used when the sample sizes are all the same. Two-Way Analysis of Variance An extension to the one-way analysis of variance. There are two independent variables. There are three sets of hypothesis with the two-way ANOVA. The first null hypothesis is that there is no interaction between the two factors. The second null hypothesis is that the population means of the first factor are equal. The third null hypothesis is that the population means of the second factor are equal. Factors The two independent variables in a two-way ANOVA. Treatment Groups Groups formed by making all possible combinations of the two factors. For example, if the first factor has 3 levels and the second factor has 2 levels, then there will be 3x2=6 different treatment groups. Interaction Effect

The effect one factor has on the other factor Main Effect The effects of the independent variables.

Stats: F-Test

The F-distribution is formed by the ratio of two independent chisquare variables divided by their respective degrees of freedom. Since F is formed by chi-square, many of the chi-square properties carry over to the F distribution.     

The F-values are all non-negative The distribution is non-symmetric The mean is approximately 1 There are two independent degrees of freedom, one for the numerator, and one for the denominator. There are many different F distributions, one for each pair of degrees of freedom.

F-Test The F-test is designed to test if two population variances are equal. It does this by comparing the ratio of two variances. So, if the variances are equal, the ratio of the variances will be 1. All hypothesis testing is done under the assumption the null hypothesis is true

If the null hypothesis is true, then the F test-statistic given above can be simplified (dramatically). This ratio of sample variances will be test statistic used. If the null hypothesis is false, then we will reject the null hypothesis that the ratio was equal to 1 and our assumption that they were equal.

There are several different F-tables. Each one has a different level of significance. So, find the correct level of significance first, and then look up the numerator degrees of freedom and the denominator degrees of freedom to find the critical value. You will notice that all of the tables only give level of significance for right tail tests. Because the F distribution is not symmetric, and there are no negative values, you may not simply take the opposite of the right critical value to find the left critical value. The way to find a left critical value is to reverse the degrees of freedom, look up the right critical value, and then take the reciprocal of this value. For example, the critical value with 0.05 on the left with 12 numerator and 15 denominator degrees of freedom is found of taking the reciprocal of the critical value with 0.05 on the right with 15 numerator and 12 denominator degrees of freedom. Avoiding Left Critical Values

Since the left critical values are a pain to calculate, they are often avoided altogether. This is the procedure followed in the textbook. You can force the F test into a right tail test by placing the sample with the large variance in the numerator and the smaller variance in the denominator. It does not matter which sample has the larger sample size, only which sample has the larger variance. The numerator degrees of freedom will be the degrees of freedom for whichever sample has the larger variance (since it is in the numerator) and the denominator degrees of freedom will be the degrees of freedom for whichever sample has the smaller variance (since it is in the denominator). If a two-tail test is being conducted, you still have to divide alpha by 2, but you only look up and compare the right critical value. Assumptions / Notes     

 

The larger variance should always be placed in the numerator The test statistic is F = s1^2 / s2^2 where s1^2 > s2^2 Divide alpha by 2 for a two tail test and then find the right critical value If standard deviations are given instead of variances, they must be squared When the degrees of freedom aren't given in the table, go with the value with the larger critical value (this happens to be the smaller degrees of freedom). This is so that you are less likely to reject in error (type I error) The populations from which the samples were obtained must be normal. The samples must be independent

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Stats: One-Way ANOVA A One-Way Analysis of Variance is a way to test the equality of three or more means at one time by using variances. Assumptions   

The populations from which the samples were obtained must be normally or approximately normally distributed. The samples must be independent. The variances of the populations must be equal.

Hypotheses

The null hypothesis will be that all population means are equal, the alternative hypothesis is that at least one mean is different. In the following, lower case letters apply to the individual samples and capital letters apply to the entire set collectively. That is, n is one of many sample sizes, but N is the total sample size. Grand Mean

The grand mean of a set of samples is the total of all the data values divided by the total sample size. This requires that you have all of the sample data available to you, which is usually the case, but not always. It turns out that all that is necessary to find perform a one-way analysis of variance are the number of samples, the sample means, the sample variances, and the sample sizes. Another way to find the grand mean is to find the weighted average of the sample means. The weight applied is the sample size.

Total Variation

The total variation (not variance) is comprised the sum of the squares of the differences of each mean with the grand mean. There is the between group variation and the within group variation. The whole idea behind the analysis of variance is to compare the ratio of between group variance to within group variance. If the variance caused by the interaction between the samples is much larger when compared to the variance that appears within each group, then it is because the means aren't the same. Between Group Variation

The variation due to the interaction between the samples is denoted SS(B) for Sum of Squares Between groups. If the sample means are close to each other (and therefore the Grand Mean) this will be small. There are k samples involved with one data value for each sample (the sample mean), so there are k-1 degrees of freedom. The variance due to the interaction between the samples is denoted MS(B) for Mean Square Between groups. This is the between group variation divided by its degrees of freedom. It is also denoted by

.

Within Group Variation

The variation due to differences within individual samples, denoted SS(W) for Sum of Squares Within groups. Each sample is considered independently, no interaction between samples is involved. The degrees of freedom is equal to the sum of the individual degrees of freedom for each sample. Since each sample has degrees of freedom equal to one less than their sample sizes, and there are k samples, the total degrees of freedom is k less than the total sample size: df = N - k. The variance due to the differences within individual samples is denoted MS(W) for Mean Square Within groups. This is the within group variation divided by its degrees of freedom. It is also denoted by . It is the weighted average of the variances (weighted with the degrees of freedom).

F test statistic

Recall that a F variable is the ratio of two independent chi-square variables divided by their respective degrees of freedom. Also recall that the F test statistic is the ratio of two sample variances, well, it turns out that's exactly what we have here. The F test statistic is found by dividing the between group variance by the within group variance. The degrees of freedom for the numerator are the degrees of freedom for the between group (k-1) and the degrees of freedom for the denominator are the degrees of freedom for the within group (N-k). Summary Table

All of this sounds like a lot to , and it is. However, there is a table which makes things really nice. SS

df

MS

F

Between

SS(B)

k-1

SS(B) ----------k-1

MS(B) -------------MS(W)

Within

SS(W)

N-k

SS(W) ----------N-k

.

Total

SS(W) + SS(B)

N-1

.

.

Notice that each Mean Square is just the Sum of Squares divided by its degrees of freedom, and the F value is the ratio of the mean squares. Do not put the largest variance in the numerator, always divide the between variance by the within variance. If the between variance is smaller than the within variance, then the means are really close to each other and you will fail to reject the claim that they are all equal. The degrees of freedom of the F-test are in the same order they appear in the table (nifty, eh?). Decision Rule

The decision will be to reject the null hypothesis if the test statistic from the table is greater than the F critical value with k-1 numerator and N-k denominator degrees of freedom.

If the decision is to reject the null, then at least one of the means is different. However, the ANOVA does not tell you where the difference lies. For this, you need another test, either the Scheffe' or Tukey test. TI-82

Ok, now for the really good news. There's a program called ANOVA for the TI-82 calculator which will do all of the calculations and give you the values that go into the table for you. You must have the sample means, sample variances, and sample sizes to use the program. If you have the sum of squares, then it is much easier to finish the table by hand (this is what we'll do with the two-way analysis of variance)

Table of Contents

ANOVA Program Performs a one-way Analysis of Variance. List 1 must contain the means of the samples, list 2 must contain the sample variances, and list 3 must contain the sample sizes. Note that the three lists must be the same size. The is reminded of these requirements when running the program. The grand mean is displayed, followed by the sum of squares, degrees of freedom, and mean sum of squares for the between group and within group. The total sum of squares and degrees of freedom, along with the F test statistic is also shown. Upon completion, the program will give the the chance to run the Scheffe test if the sample sizes are different or the Tukey test if the sample sizes are the same. All possible pairs are compared.

Stats: Scheffe' and Tukey Tests

When the decision from the One-Way Analysis of Variance is to reject the null hypothesis, it means that at least one of the means isn't the same as the other means. What we need is a way to figure out where the differences lie, not just that there is a difference. This is where the Scheffe' and Tukey tests come into play. They will help us analyze pairs of means to see if there is a difference -- much like the difference of two means covered earlier.

Hypotheses Both tests are set up to test if pairs of means are different. The formulas refer to mean i and mean j. The values of i and j vary, and the total number of tests will be equal to a combination of k objects, 2 at a time C(k,2), where k is the number of samples.

Scheffé Test The Scheffe' test is customarily used with unequal sample sizes, although it could be used with equal sample sizes. The critical value for the Scheffe' test is the degrees of freedom for the between variance times the critical value for the one-way ANOVA. This simplifies to be: CV = (k-1) F(k-1,N-k,alpha)

The test statistic is a little bit harder to compute. Pure mathematicians will argue that this shouldn't be called F because it doesn't have an F distribution (it's the degrees of freedom times an F), but we'll live it with it. Reject H0 if the test statistic is greater than the critical value. Note, this is a right tail test. If there is no difference between the means, the numerator will be close to zero, and so performing a left tail test wouldn't show anything.

Tukey Test The Tukey test is only usable when the sample sizes are the same. The Critical Value is looked up in a table. It is Table N in the Bluman text. There are actually several different tables, one for each level of significance. The number of

samples, k, is used as a index along the top, and the degrees of freedom for the within group variance, v = N-k, are used as an index along the left side. The test statistic is found by dividing the difference between the means by the square root of the ratio of the within group variation and the sample size. Reject the null hypothesis if the absolute value of the test statistic is greater than the critical value (just like the linear correlation coefficient critical values).

TI-82 The ANOVA program for the TI-82 will do all of the pairwise comparisons for you after it has given the ANOVA summary table. You will need to know how to find the critical values and make the comparisons.

Table of Contents

Stats: Two-Way ANOVA The two-way analysis of variance is an extension to the one-way analysis of variance. There are two independent variables (hence the name two-way). Assumptions    

The populations from which the samples were obtained must be normally or approximately normally distributed. The samples must be independent. The variances of the populations must be equal. The groups must have the same sample size.

Hypotheses

There are three sets of hypothesis with the two-way ANOVA. The null hypotheses for each of the sets are given below.

1. The population means of the first factor are equal. This is like the one-way ANOVA for the row factor. 2. The population means of the second factor are equal. This is like the one-way ANOVA for the column factor. 3. There is no interaction between the two factors. This is similar to performing a test for independence with contingency tables. Factors

The two independent variables in a two-way ANOVA are called factors. The idea is that there are two variables, factors, which affect the dependent variable. Each factor will have two or more levels within it, and the degrees of freedom for each factor is one less than the number of levels. Treatment Groups

Treatement Groups are formed by making all possible combinations of the two factors. For example, if the first factor has 3 levels and the second factor has 2 levels, then there will be 3x2=6 different treatment groups. As an example, let's assume we're planting corn. The type of seed and type of fertilizer are the two factors we're considering in this example. This example has 15 treatment groups. There are 3-1=2 degrees of freedom for the type of seed, and 5-1=4 degrees of freedom for the type of fertilizer. There are 2*4 = 8 degrees of freedom for the interaction between the type of seed and type of fertilizer. The data that actually appears in the table are samples. In this case, 2 samples from each treatment group were taken. Fert I

Fert II

Fert III

Fert IV

Fert V

Seed A-402

106, 110

95, 100

94, 107

103, 104

100, 102

Seed B-894

110, 112

98, 99

100, 101

108, 112

105, 107

Seed C-952

94, 97

86, 87

98, 99

99, 101

94, 98

Main Effect

The main effect involves the independent variables one at a time. The interaction is ignored for this part. Just the rows or just the columns are used, not mixed. This is the

part which is similar to the one-way analysis of variance. Each of the variances calculated to analyze the main effects are like the between variances Interaction Effect

The interaction effect is the effect that one factor has on the other factor. The degrees of freedom here is the product of the two degrees of freedom for each factor. Within Variation

The Within variation is the sum of squares within each treatment group. You have one less than the sample size ( all treatment groups must have the same sample size for a two-way ANOVA) for each treatment group. The total number of treatment groups is the product of the number of levels for each factor. The within variance is the within variation divided by its degrees of freedom. The within group is also called the error. F-Tests

There is an F-test for each of the hypotheses, and the F-test is the mean square for each main effect and the interaction effect divided by the within variance. The numerator degrees of freedom come from each effect, and the denominator degrees of freedom is the degrees of freedom for the within variance in each case. Two-Way ANOVA Table

It is assumed that main effect A has a levels (and A = a-1 df), main effect B has b levels (and B = b-1 df), n is the sample size of each treatment, and N = abn is the total sample size. Notice the overall degrees of freedom is once again one less than the total sample size. Source

SS

df

MS

F

Main Effect A

given

A, a-1

SS / df

MS(A) / MS(W)

Main Effect B

given

B, b-1

SS / df

MS(B) / MS(W)

Interaction Effect

given

A*B, (a-1)(b-1)

SS / df

Within

given

N - ab, ab(n-1)

SS / df

sum of others

N - 1, abn - 1

Total

MS(A*B) / MS(W)

Summary

The following results are calculated using the Quattro Pro spreadsheet. It provides the p-value and the critical values are for alpha = 0.05. Source of Variation

SS

df

MS

F

P-value

F-crit

Seed

512.8667

2

256.4333

28.283

0.000008

3.682

Fertilizer

449.4667

4

112.3667

12.393

0.000119

3.056

Interaction

143.1333

8

17.8917

1.973

0.122090

2.641

Within

136.0000

15

9.0667

1241.4667

29

Total

From the above results, we can see that the main effects are both significant, but the interaction between them isn't. That is, the types of seed aren't all equal, and the types of fertilizer aren't all equal, but the type of seed doesn't interact with the type of fertilizer.

Error in Bluman Textbook

The two-way ANOVA, Example 13-9, in the Bluman text has the incorrect values in it. The student would have no way of knowing this because the book doesn't explain how to calculate the values. Here is the correct table: Source of Variation

SS

df

MS

F

Sample

3.920

1

3.920

4.752

Column

9.680

1

9.680

11.733

54.080

1

54.080

65.552

3.300

4

0.825

70.980

7

Interaction Within Total

The student will be responsible for finishing the table, not for coming up with the sum of squares which go into the table in the first place.