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.
1) Denote each voter's preference by Y, with Y= 1 if the voter prefers the incumbent and Y = 0 if the voter prefers the challenger. Y is a Bernoulli random variable with probability Pr(Y=1) = p and Pr(Y=0) = 1 - p.
Using the survey results, you can estimate p with p' = $\boxed{215}$/400 = $\boxed{0.5375}$. Hint: Write the estimate in four decimal places.
2) Using the estimator of the variance of p', p'(1 - p')/n, you can calculate the standard error of your estimator. The estimated variance of p' is $\boxed{0.5375(1 - 0.5375)/400}$ and thus, The standard error is SE(p') = $\sqrt{var(p')} = \boxed{0.025}$. Hint: Write the SE in four decimal places.
3) To conduct the hypothesis testing of Ho: p=0.5 vs. H₁: p≠0.5, you compute t-statistic, t = (p' - 0.5)/SE(p') = $\boxed{1.5}$. Hint: Write the answer in three decimal places.
4) Using the t-statistic, you find p-value = 2Φ(-|t|) = $\boxed{0.13}$. Hint: Write the answer in two decimal places.
