Question

1. Seat Belt Use ~   The U.S. Department of Transportation collected seat belt use data by stationing observers at randomly selected roadway sites and recording the number of vehicle occupants who were wearing seat belts. A random sample of 1000 vehicle occupants in the Northeast shows that 909 were wearing seat belts, while a random sample of 1000 vehicle occupants in the Midwest showed that 855 were wearing seat belts.

CREDIT ONLY GIVEN IF WORK IS SHOWN. ANSWERS WITHOUT WRITTEN WORK WILL NOT BE GRADED.

1a. Let p1​ and p2​ denote the population proportion of occupants wearing seat belts in the Northeast and Midwest, respectively. Which of the following is the appropriate set of hypotheses to test if the rate at which Northeast drivers wear seat belts is higher than those in the Midwest

A. H0​:p1​>p2​ vs. Ha​:p1​≤p2​

B. H0​:p1​−p2​=−0.054 vs. Ha​:p1​−p2​<−0.054

C. H0​:p1​=p2​ vs.Ha​:p1​<p2​

D. H0​:p^​1​=p^​2​ vs. Ha​:p^​1​<p^​2​

E. None of these is the correct set of hypotheses.

1b. The null hypothesis H0​ believes that the rate at which drivers in the Midwest and Northeast wear seat belts is the same, and that any observed difference across the two samples is attributable to random chance alone. If this was, in fact, the case, what is the best estimate of the rate at which drivers wear seat belts, regardless of whether they live in the Northeast or Midwest? Submit your answer out to four decimal places.

1c. The study results give phat^​1​−phat^​2​ of -0.054. Approximately how far off, on average, would we expect this sample statistic to be from the parameter of interest, p1​−p2​? Submit your answer out to four decimals.

1d. Which of the following is a correct interpretation of the quantity you computed in Q3?

A In repeated samples, the quantity in Q3 tells us how often we would expect our estimate of \hat{p}_2-\hat{p}_2p^​2​−p^​2​ to be equal to 0 approximately.

B Since this quantity is less than 0.05, we will reject the null hypothesis.

C The quantity in Q3 tells us, over repeated samples, how much we would expect our estimate to be, on average, away from the true population difference in proportions.

D The quantity in Q3 tells us how often any estimate of the difference in the population proportions will be wrong.

1e. Compute the test statistic and p-value associated with these hypotheses. Submit your answers for both out to at least six decimal places.

1f. Using the p-value from Q5 and a significance level of α = 0.05, what do you conclude?

A i. There is insufficient evidence to reject the null hypothesis. It appears the rate is higher in the Northeast.

B ii. There is sufficient evidence to reject the null hypothesis. It appears the rate is higher in the Northeast.

C iii. There is insufficient evidence to reject the null hypothesis. It appears the rate is not higher in the Northeast.

D iv. There is sufficient evidence to reject the null hypothesis. It appears the rate is not higher in the Northeast.

Answer 1