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Part B: Analytical questions (80 marks)

Instruction: Answer all questions and please show all the steps in detail.

Question 21 (6 marks)

Read the following and answer the questions.

A number of automobile accidents occur at various high-risk intersections in Central despite traffic lights. Traffic department wants to know if automobile accident is a very serious problem in Central compared to other districts in Hong Kong. Some intersections in Central have been chosen. The number of accidents at those intersections has been recorded for three months.

a. Is the data collected in the above case qualitative or quantitative data?

b. Is the data collected in the above case discrete or continuous data?

c. What is the level of measurement for the data collected in the above case?

d. What statistic would you expect to provide the most representative measure of central

tendency for the data collected in the above case?

e. What statistic would you expect to provide the most representative measure of dispersion for

the data collected in the above case?

f. If you want to know if automobile accident is a very serious problem in Central compared to

other districts in Hong Kong, what statistic test would you use?

Question 22 (7 marks)

Read the following and answer the questions.

Traffic department wants to know if automobile accident is a very serious problem in Central compared to other districts in Hong Kong. Nine intersections in Central have been chosen at random. The number of accidents at those intersections has been recorded for three months. The average number of accidents at an intersection in Hong Kong in the past three months is 3.

The number of accidents at the chosen intersections in Central:

a. Calculate the mean number of accidents in Central.

b. Calculate the sample standard deviation for the number of accidents in Central.

c. State the null hypothesis if traffic department wants to test whether automobile accident is a

very serious problem in Central compared to other districts in Hong Kong.

d. Continue from part c, state the corresponding alternative hypothesis of the test.

e. Continue from part c, calculate the test statistics of the test.

f. Continue from part e, what is the critical value of the test if the significant level is 5%?

g. According to the test, is automobile accident a very serious problem in Central compared to

other districts in Hong Kong?

Question 23 (5 marks)

A team of experts claims that a modification in the type of light will reduce these accidents. The lights at the chosen intersections in Central were modified. The number of accidents before and after the modification has been recorded.

a. State the null hypothesis if traffic department wants to test whether the modification reduces the number of traffic accidents in Central.

b. Continue from part a, state the corresponding alternative hypothesis of the test.

c. Continue from part a, calculate the test statistics of the test.

d. Continue from part c, what is the critical value of the test if the significant level is 1%?

e. According to the test, is it reasonable to conclude that the modification reduced the number

of traffic accidents?

Question 24 (15 marks)

Answer the following questions.

a. The number of orders that come into the sales office of ABC LTD each month is normally

distributed with a mean of 600 and a standard deviation of 100. Approximately less than

how many orders does the sales office receive such that the probability is 0.6

b. The number of orders that come into the sales office of DEF LTD each month is normally distributed with a mean of 500. Suppose the probability that the sales office receives more

than 692 orders is approximately 0.1. Please calculate the standard deviation of the

distribution of the orders.

c. The number of orders that come into the sales office of XYZ LTD each month is normally

distributed with a standard deviation of 200. Suppose the probability that the sales office receives less than 794 orders is approximately 0.15. Please calculate the mean of the distribution of the orders.

Answer the following questions.

a. The length of time it takes to fill an order at a ABC sandwich shop is normally distributed

with a mean of 6 minutes and a standard deviation of 2 minute. What is the probability that the average waiting time for a random sample of 100 customers is between 5.6 and 6 minutes?

b. The length of time it takes to fill an order at a DEF sandwich shop is normally distributed with a mean of 5 minutes and a standard deviation of 2 minute. Suppose the probability that the average waiting time for a random sample of customers between 4.75 and 5 minutes is 0.3413. What is the number of customers in the sample?

c. The length of time it takes to fill an order at a XYZ sandwich shop is normally distributed with a mean of 7 minutes. Suppose the probability that the average waiting time for a random sample of 36 customers between 6.8 and 7.2 minutes is 0.383. What is the standard deviation of the waiting time?

Question 26 (10 marks)

A random sample of 16 people spent an average of 45 minutes driving to work.

a. Assume the population standard deviation (𝜎) is known and equals 8, find a 95% confidence

interval for the population mean (μ)

b. Assume the population standard deviation (𝜎) is unknown and the sample standard deviation

(s) equals 8, find a 95% confidence interval for the population mean (μ)

Question 27 (10 marks)

Answer the following questions.

a. The mean length of a work week for the population of workers was reported to be 45 hours.

Suppose that we would like to take a current sample of workers to see whether the mean length of a work week has decreased from the previously reported 45 hours. Suppose a current sample of 100 workers provided a sample mean (𝑥̅) of 42 hours and sample standard deviation (s) of 10 hours. Use 5% significance level to test whether the mean length of a work week has decreased from the previously reported 45 hours.

b. You have been asked to determine if male and female managers in a big company have the same level of education. Independent random samples were taken of 100 male managers and 100 female managers. For male managers, the sample mean was 16 years of schooling with sample standard deviation (s1) equals to 3. For female managers, the sample mean was 14 years of schooling with sample standard deviation (s2) equals to 4. Do you have any reason at the 5% significance level to think that male and female managers have the same level of education?

STAT A221F (1904) Page 8 of 8

Question 28 (12 marks)

Answer the following questions.

a. The manager of a company wishes to study the number of hours the employees spend at

their desktop computers. The manager selected a sample of six employees from three teams: A, B and C. At the 0.05 significance level, can the manager conclude there is a difference in the mean number of hours spent per week by team? Given the treatment sum of squares is 10 and total sum of squares is 40.

b. A real estate developer is considering investing in a shopping mall. Three areas are being evaluated. Of particular importance is the income in the area surrounding the proposed mall. A random sample of five families is selected near each proposed mall. At the 0.05 significance level, can the developer conclude there is difference in the mean income? Given the treatment sum of squares is 44 and total sum of squares is 80.

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