In a survey of 500 likely voters, 271 responded that they would vote for the incumbent and 229 responded that they would vote for the challenger. Let pp denote the fraction of all likely voters who preferred the incumbent at the time of the survey, and let p̂ p^ be the fraction of survey respondents who preferred the incumbent.

a. Use the survey results to estimate pp.

b. Test the hypothesis H0:p=0.5 vs. Ha:p≠0.5H0:p=0.5 vs. Ha:p≠0.5 at the 5% significance level. Remember to state your assumptions. Keep in mind that you should calculate the standard error assuming the null hypothesis is true. This means that you should use pp in your standard error calculations.

c. What is the p-value for the test H0:p=0.5 vs. Ha:p≠0.5H0:p=0.5 vs. Ha:p≠0.5?

d. What is the p-value for the test H0:p=0.5 vs. Ha:p>0.5H0:p=0.5 vs. Ha:p>0.5?

e. Do the results from (c) to (d) differ? Why or why not?

f. Did the survey contain statistically significant evidence that the incumbent was ahead of the challenger at the time of the survey? Explain.

g. Suppose you knew (you are now omniscient) that the fraction of all likely voters who preferred the incumbent at the time of the survey was p=0.5p=0.5. You would like to do a survey of all likely voters, but can only conduct a survey on landlines, i.e. cell phones are not included. A survey using a simple random sample of 500 landline telephone numbers finds that 54% of respondents support the incumbent. Is there evidence that the survey is biased? Explain.

h. A survey using a simple random sample of 500 landline telephone numbers finds that 55% of respondents support the incumbent. Is there evidence that the survey is biased? Explain. Compare to part (g) and explain any similarities and differences in conclusions.

i. For a sample of 500, what proportion of respondents that support the incumbent should there be to not worry about bias?