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A survey of 1055 registered voters is conducted, and the voters are asked to choose between candidate A and candidate B. Let p denote the fraction of voters in the population who prefer candidate A, and let denote the fraction of voters in the sample who prefer Candidate A.

(a) You are interested in the competing hypotheses H0: p = 0.5 vs.

H1. p ≠ 0.5. Suppose that you decide to reject H0 if | - 0.51 > 0.02.

i. What is the size of this test?

ii. Compute the power of this test if p = 0.53.

(b) In the survey, = 0.54.

i. Test H0: p = 0.5 vs. H1. p ≠ 0.5 using a 5% significance level.

ii. Test H0: p = 0.5 vs. H1: p > 0.5 using a 5% significance level.

iii. Construct a 95% confidence interval for p.

iv. Construct a 99% confidence interval for p.

v. Construct a 50% confidence interval for p.

(c) Suppose that the survey is carried out 20 times, using independently selected voters in each survey. For each of these 20 surveys, a 95% confidence interval for p is constructed.

i. What is the probability that the true value of p is contained in all 20 of these confidence intervals?

ii. How many of these confidence intervals do you expect to contain the true value of p?

(d) In survey jargon, the "margin of error'' is 1.96 × SE(); that is, it is half the length of 95% confidence interval. Suppose you wanted to design a survey that had a margin of error of at most l%. That is, you wanted Pr( - p | > 0.01) ≤ 0.05. How large should n be if the survey uses simple random sampling?

(a) You are interested in the competing hypotheses H0: p = 0.5 vs.

H1. p ≠ 0.5. Suppose that you decide to reject H0 if | - 0.51 > 0.02.

i. What is the size of this test?

ii. Compute the power of this test if p = 0.53.

(b) In the survey, = 0.54.

i. Test H0: p = 0.5 vs. H1. p ≠ 0.5 using a 5% significance level.

ii. Test H0: p = 0.5 vs. H1: p > 0.5 using a 5% significance level.

iii. Construct a 95% confidence interval for p.

iv. Construct a 99% confidence interval for p.

v. Construct a 50% confidence interval for p.

(c) Suppose that the survey is carried out 20 times, using independently selected voters in each survey. For each of these 20 surveys, a 95% confidence interval for p is constructed.

i. What is the probability that the true value of p is contained in all 20 of these confidence intervals?

ii. How many of these confidence intervals do you expect to contain the true value of p?

(d) In survey jargon, the "margin of error'' is 1.96 × SE(); that is, it is half the length of 95% confidence interval. Suppose you wanted to design a survey that had a margin of error of at most l%. That is, you wanted Pr( - p | > 0.01) ≤ 0.05. How large should n be if the survey uses simple random sampling?

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