1. A drug manufacturer is developing a vaccine against a particular viral infection. In a preliminary trial, it administered the vaccine to 350 volunteers, 320 of whom developed an immunity to the infection. If the manufacturer can conclude that the vaccine is more than 90% effective, it will proceed to the next stage of development.

a. Define the population and the parameter of interest in this problem.

b. Based on the data obtained, should the company proceed to the next stage of development? Clearly state the null and alternative hypotheses in terms of the parameter of interest, calculate the test statistic and the p-value, and state your conclusion in the context of the problem. Use the 5% significance level.

c. Suppose that it’s later discovered that the vaccine is more crucial for public health than had been initially assessed, as the disease it is meant to prevent is deadlier than thought. In light of this, what should the company do regarding the significance level of the test, raise it, lower it, or keep it unchanged? Explain your answer.

2. The delivery time (in business days) for international parcel by post for a random sample of 26 parcels are summarized as follows:

n 26

Mean 9.46

Median 8

SD 5.38

a. Based on these data, are we willing to conclude that on average an international mail takes less than than 12 business days? Clearly define the parameter of interest, formulate the null and  alternative hypotheses, calculate the test statistic and p-value, and state your conclusion in the context of the problem. Use the 10% significance level. (The standard deviation of delivery time is known to be σ = 2.70 days)

b. Calculate the 85% Confidence Interval for the average delivery time, and explain in plain English what it means in the context of the problem. (In your calculation, use the population standard deviation σ = 4.70).

c. In both the Hypothesis Test and Confidence Interval above, do we have to assume that the distribution of delivery time is Normal? If so, is it a reasonable assumption to make? Briefly justify your answer.

3. When the COVID 19 pandemic started and there was a lockdown order, about 54% of workers reported to actually prefer working from home. Six months later, in a survey of 150 workers 77 said they prefer working from home.

a. Based on this survey, can we conclude that the proportion of workers who actually prefer working from home has changed? Test at the 5% significance level. Be sure to clearly define the  parameter of interest, the null and alternative hypotheses, report the test statistic and the p-value, state your conclusion in the context of the problem.

b. What does committing Type I Error entails in this question?

4. The time needed for college students to complete a certain paper-and-pencil maze follows a Normal distribution with a mean of 30 seconds and a standard deviation of 2.6 seconds. You wish to see if the mean time mu is changed by vigorous exercise, so you have a group of 20 college students exercise vigorously for 30 minutes and then complete the maze. It takes them an average of ¯x=28.8 seconds.

a. Calculate the Standard Error associated with the sample mean, and explain what the number means in the context of the problem.

b. Based on these data can we conclude that average time to complete the maze changes after vigorous exercise? Define the parameter of interest, state the null and alternative hypotheses,  calculate the test statistic and p-value, and state your conclusion in the context of the problem. Use the 1% significance level.

c. Interpret in plain English and in the context of the problem what the p-value you calculated above mean.