In a survey of 400 likely voters, 215 responded that they would vote for the incumbent and 185 responded that they would vote for the challenger. Let p denote the fraction of all likely voters who preferred the incumbent at the time of the survey, and let P^ be the fraction of survey respondents who preferred the incumbent.

a, Use the survey results to estimate P.

b, Use the extimator of the variance of P^, P^(1-P^)/n, to calculate the standard error of your estimator.

c. what is the p-value for the test H0: P=0.5 vs. H1: P not =0.5?

d, what is the p-value for the test H0: p=0.5 vs. H1: p larger than 0.5?

e. why do the results from c and d differ?

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