Now suppose the experiment is conducted using a sample size

Question

Questions 1 through 3 rely on the following information:

Suppose the state legislators in a particular state are concerned about auto fatalities in the state. Current law in the state sets the maximum speed limit on highways at 65 mph. The legislators are considering low sing the maximum speed limit to 55 mph. To estimate whether or not this change in law will affect auto fatalities, a consulting company is hired. The company recommends drawing a random sample of 10-mile sections of randomly selected highways. For each of these sections of highway, the consultants propose recording the number of auto accidents within the highway section that results in a fatality over a six month period of time, while the speed limit is still 65 mph, and then recording the number of auto accidents resulting in a fatality over another six month period of time after the speed limit has been mimed to 55 mph. Let X_i^b denote the number of fatalities on the ith stitch of road when the speed limit is 65 mph, and let X_i^a denote the number of fatalities on that same segment of road after the speed limit has been reduced to 55 mph. Prior to this "experiment" being conducted, X_i^a and X_i^b can both be viewed as random variables, and, thus, so can their difference Y_i = X_i^a - X_i^b. The consultants recommend n sections of highway be randomly selected. Hence, they will draw a random sample of n differences Y₁, Y₂, . . . Y_n.

1. Eventually, the consultants will conduct a test. The null hypothesis is that reducing the speed limit from 65 mph to 55 mph will have no effect on the number of auto fatalities, while the alternative hypothesis is that there will be a decline in the number of fatal accidents. State the null hypothesis (H₀) and alternative hypothesis (H₁) formally in term of μ.

2. Now, suppose the "experiment" is conducted. Using a sample size of n = 800, the consulting company obtains estimates of y^- = -.11 and s = 1.03. What is the value of the t statistic for testing H₀ versus H₁?

3. Two or false. We cannot reject H₀ at the 1 percent level.

Questions 4 through 6 rely on the following information:

In a campaign involving two candidates, Candidate A and Candidate B, one of the candidates, Candidate A, hires you to conduct a poll. For this poll, you plan on asking respondents "Will you vote for Candidate A?" Let Y_i = 1 if the ith respondent says s/he will vote for A and let Y, = 0 otherwise. (Note that Y_i is a Bernoulli random variable.) When you conduct the poll, if you choose respondents randomly, you are drawing a random sample Y₁, . . . , Yn. Let θ denote the fraction of the voting public that supports A at the time the poll is taken.

4. Candidate A is obviously interested in θ, because he is interested in winning the election. Formally state the null hypothesis (H₀), in terms of θ, that he will not win the election. Also, formally state the alternative hypothesis (H₁), in terms of θ, that he out win the election.

5. Suppose you draw a random sample of sire n = 1,200, and you get y^- = .52. (Note that se(y^-) =  √(y^-(1 -y^-)/n).) Use this information to test H₀, versus H₁, at the 10 percent significance level.

6. Give the 95 percent confidence interval for θ.