(Solution) - In a survey of400 likely voters 215 responded that they -(2025 Original AI-Free Solution)

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In a survey of400 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 estimator of the variance of , (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 ? 0.5?
(d) What is the p-value for the test H0: p = 0.5 vs. H1 p > 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.