Exercise 1 (26 marks = 10 + 10 + 6)

The management of a large fleet of vehicles is concerned about vehicle downtime, i.e., about the number of days a vehicle is unavailable due to some repair after an accident, and decided to experiment with a new bumper that might reduce the damage in low-speed collisions. In order to see whether this new bumper is indeed effective, she took a random sample of the company’s cars and had the new bumper installed on them for a trial period of two months. At the end of the trial period, accident data showed that the cars with the new bumper had 12 repair incidents, while during the same period another random sample of cars equipped with the original bumper had 9 repair incidents. Vehicle downtime due to these repair incidents are shown in the following table.

In this table New and Original denote vehicle downtime in number of days per repair incident for cars with the new bumper and with the original bumper, respectively.

Using this data set perform the following tasks. Do all calculations ‘manually’ (i.e., with a hand calculator without using R, Excel, or any other software), showing the relevant formulas and the major steps.

(a) (10 marks)

Do these data provide sufficient information to indicate at the 5% significance level that the new bumper reduces the variance of vehicle downtime? Briefly discuss the steps of the test you perform showing all relevant details.

(b) (10 marks)

Do these data provide sufficient information to indicate at the 5% significance level that the new bumper reduces mean vehicle downtime? Briefly discuss the steps of the test you perform showing all relevant details.

(c) (6 marks)

What are the requirements of the hypothesis tests you performed in parts (a) and (b)? Are the requirements about the experimental design, the variable of interest and the measurement scale met? Explain your answers.

Exercise 2 (38 marks = 12 + 14 + 12)

To make children’s toys safer and more difficult to change, toy packaging has become cumbersome for parents to remove in many cases. Accordingly, the director of marketing at Toys4Tots, a large toy manufacturer, wants to evaluate the effectiveness of a new packaging design that engineers claim will reduce customer complaints. Customer satisfaction surveys were sent to 250 randomly selected parents who registered toys packaged under the old design and 250 randomly selected parents who registered toys packaged under the new design. Of these, 85 parents expressed dissatisfaction with packaging of the old design, and 70 parents expressed dissatisfaction with packaging of the new design. Suppose that the director charges you to test whether the new packaging design can be indeed expected to reduce customer complaints.

(a) (12 marks)

Perform the appropriate test with R at the 5% significance level. State the  hypotheses, report the observed test statistic, make a statistical decision, and draw the implied conclusion.

(b) (14 marks)

Suppose now that you have access neither to R nor to a computer, so you can only perform the test by manual calculations. Briefly discuss the steps and show all relevant details of your calculations. Is your observed test statistic the same as the one reported by R in part (a)? Do you arrive at the same decision and  conclusion as in part (a)?

(c) (12 marks)

What conditions are required by the tests you performed in parts (a) and (b)? Are they satisfied this time? Provide as many supporting evidence as possible.

Exercise 3 (36 marks = 10 + 14 + 4 + 8)

Ten patients suffering from arthritis take part in an experiment to evaluate the relative effectiveness of three pain-relieving drugs. The following table shows how they ranked the three drugs in order of effectiveness (1: least effective, 3: most effective).

This data is also saved in the a2e3.xlsx file.

(a) (10 marks)

What is the variable of interest? Is it qualitative or quantitative? If it is qualitative, is it ranked or unranked? If it is quantitative, is it discrete or continuous? What is its level of measurement? What is the experimental design? Explain your answers.