Poisson 6d5p

BUSINESS STATISTICS REVISION -MOCK QUESTIONS SECTION A:

A1.

A distribution of 6 scores has a median of 21. If the highest score increases 3 points, the median will become. (A) (B) (C) (D)

A2.

the population mean exceeds 3.2 the distribution is bell-shaped the distribution is skewed positively the distribution is skewed negatively

One card is drawn from a standard 52 card deck. In describing the occurrence of two possible events, an Ace and a King, these two events are said to be _____. (A) (B) (C) (D)

A5.

There is an error in the calculation. The sample size is 1. All the values in the population are 28. The range is 28.

If the mean, median and mode of a distribution are 5, 6, 7 respectively, then the distribution is _____. (A) (B) (C) (D)

A4.

25 24 21.5 21

If you are told a population has a mean of 28 and a variance of 0, what must you conclude? (A) (B) (C) (D)

A3.

There are 10 multiple-choice questions in this section. Each question is worth 2 marks.

independent mutually exclusive random variables randomly independent

One hundred students took a test on which the mean score was 73 with a variance of 64. A grade of ‘A’ was given to all who scored 85 or better. Approximately how many ‘A’ were there, assuming scores were normally distributed? (Choose the closest.) (A) (B) (C) (D)

20 17 7 1

A6.

If Z is a standard normal variable, then the area to the left of Z = 0.65 is _____. (A) (B) (C) (D)

A7.

Z-scores provide information about the location of raw scores _____. (A) (B) (C) (D)

A8.

below the mean in units of the range of the distribution above the mean in units of the standard deviation of the distribution above and below the mean in units of the range of the distribution above and below the mean in standard deviation units from the mean

A t-distribution with 30 degree of freedom (df) is most similar to a _____ distribution. (A) (B) (C) (D)

A9.

0.35 0.45 0.74 0.83

normal distribution with mean = 1 and variance = 1 normal distribution with mean = 0 and variance = 30 normal distribution with mean = 0 and variance = 1 normal distribution with mean = 30 and variance = 1

What type of decision is reached when the calculated value of any test statistic falls in the critical region when a false null hypothesis is being tested? (A) (B) (C) (D)

A correct decision Type I error Type II error The type of decision cannot be determined from the information given.

A10. If the correlation between age of a car and money spent for repairs is +.90, which of the following statement is true? (A) 90% of the money spent for repairs is explained by the age of the car. (B) 90% of the money spent for repairs is unexplained by the age of the car. (C) 81% of the variation in the money spent for repairs is explained by the age of the car. (D) 81% of the variation in the money spent for repairs is unexplained by the age of the car.

SECTION B: There are 3 short questions in this section. Each question is worth 10 marks. B1.

Each member of a sample of 3,750 women aged 30-39 was measured for unaided distance vision in both right and left eyes. The results are presented in the following Table B1.

Highest Grade Right Eye

Highest Grade 750

Left Eye Second Third Grade Grade 130 60

Lowest Grade 35

Second Grade

125

775

210

40

Third Grade

60

180

885

100

Lowest Grade

20

40

90

250

Table B1 For a randomly selected person from the population sampled above: a. Compute the probability that the left eye will fall into the third grade of unaided distance vision. [1 mark] b. Compute the probability that the left eye will fall into the highest grade, given that the right eye has the lowest grade. [3 marks] c. Compute the probability that the right eye will fall into third grade and lower given that the left eye has at least second grade. [3 marks] d. Compute the probability that the right eye or left eye will fall into at least second grade [3 marks] B2.

a. If X follows a normal distribution with mean, 10, and variance, 9, compute their probabilities: i. P(X ≤ 15.13)

[2 marks]

ii. P(X ≥ 5.62)

[2 marks]

iii. P(7.96 ≤ X ≤ 9.04)

[3 marks]

b. If Z follows a standard normal distribution, find the value of a if P(Z ≥ a) = 0.96562. [3 marks]

B3.

a. Alfred, a modern farmer, planted a row of hybrid turnips as an experiment, and the resulting crop had 15 turnips. He placed these in a cold spot in his barn in November, and he plans to take them all out in April. He’s aware that some turnips may have turned rotten over this time period and the probability of it rotting is 0.2. i. What is the probability that at most 20% of the turnips rot over the period November to April? [2 marks] ii. What is the probability that between 60% to 80% of the turnips will not rot over the period November to April? [3 marks] b. Assuming that on average cars arrive at a petrol station along East Coast Road at a rate of 36 cars per hour and that a Poisson model is appropriate. i. What is the probability of more than 2 cars arriving at the petrol station in a 5 minute period? [2 marks] ii. What is the probability of between 6 to 10 cars arriving at the petrol station in a 10-minute period? [3 marks]

SECTION C: There are 2 long questions in this section. Each question is worth 25 marks. C1.

a. In a survey study on the incidence of depression in a population of psychiatric hospital s, scores on a depression measure known as DEPScore were obtained from 81 respondents. DEPScore’s methodologies combines intrinsic and extrinsic responses to arrive at a final score. Based on the responses from 81 respondents, the observed mean score was 62, with a standard deviation of 16. i. What is the 95 percent confidence interval for DEPScore?

[10 marks]

ii. DEPScore of more than 60 indicates a serious depression condition that warrant immediate treatment. The hospital senior management claims that the survey result signals a serious depression condition among the psychiatric hospital s. State the null and alternate hypothesis. [2 marks] iii. Using the survey result, perform a hypothesis test to test the hospital senior management’s claim that the DEPScore is above 60, at the 0.05 significance level. [8 marks] b. An Epidemiologist studied a randomly selected group of 25 individuals (men and women) between 30- 49 years of age and finds that their mean heart rate is 70 beats per minute. The population standard deviation is 10 beats per minute. What proportion of samples will have mean values of 74 beats per minute or greater, if repeated samples of 25 individuals are randomly selected from the population? [5 marks]

C2.

a. Distinguish between the Type I and Type II statistical error. Which error is more serious? [6 marks] b. Established in 1981, Orient Star Hotel (OSH) is a 300-room business hotel situated in the heart of Marina Bay district. It has a lounge for 250 on its premises. Although the room occupancy rate has been good, the lounge business has been relatively weak as compared to its competitors. The OSH has therefore recruited you as business development manager for the lounge. Your immediate focus is to enhance the business in the lounge and in particular, increase the beverage sales. You decide to investigate the relationship between the monthly sales on beverages and food and identify if food sales is an influencing factor. The previous ten month sales data is as shown in Table C2. Month July’13 June’13 May’13 April’13 March’13 February’13 January’13 December’12 November’12 October’12

Monthly Beverage Sales Monthly Food Sales (in $ Thousands) (in $ Thousands) 17.8 1.45 22.2 2.48 15.29 2.49 18.3 2.46 12.86 2.18 13.93 3.58 16.74 3.01 22.67 3.27 23.77 3.22 24.68 3.44 Table C2

i. Identify the independent (X) and dependent variables (Y).

[2 marks]

2 2 ii. Draw a table and compute  x,  x ,  y,  y  xy, x and y .

[7 marks]

iii. Determine the linear regression equation: yˆ = a + bx.

[6 marks]

iv. What portion of variation in monthly beverage sales is explained by the monthly food sales? [4 marks] END OF PAPER

FORMULA SHEET Binomial Distribution: n! P( x)   x (1   ) n  x x!(n  x)! Poisson Distribution:  x e  P( x)  x! Confidence Interval Estimates for the Mean using Z-distribution:



x  z / 2

n

Confidence Interval Estimates for the Mean using t-distribution: s x  t / 2,n1 n Z-distribution used as test statistic: x z / n t-distribution used as test statistic: x t s/ n Method of Least Squares:

yˆ  a  bx b

n( xy )   x  y 

a

 y  b  x  y  bx

r





n  x 2   x 

n

2

n n( xy ) ( x)( y )

n(  x 2 )  (  x ) 2 n(  y 2 )  (  y ) 2