Question

In a survey of 400 likely voter, 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.

Use the survey results to estimate p.
Use the estimator of the variance of p^ (given byp^(1−p^)/n ) to calculate the standard error of your estimator.
Construct a 95% confidence interval for p
Construct a 99% confidence interval for p
Explain why the 99% interval is wider than the 95% one.

Answer 1

To estimate p, we calculate p hat using the survey results. The standard error of the estimator is then calculated using the given formula. A 95% confidence interval is constructed using the p hat value and the standard error. The 99% confidence interval is wider because it uses a larger critical value.

Step-by-step explanation:

To estimate the fraction of all likely voters who preferred the incumbent, we can use the fraction from the survey respondents who preferred the incumbent. In this case, p hat = 215/400 = 0.5375.

To calculate the standard error of the estimator, we can use the formula for the estimator of the variance: SE(p hat) = sqrt(p hat(1 - p hat) / n). Plugging in the values, we get SE(p hat) = sqrt(0.5375(1 - 0.5375) / 400) = 0.0249.

For a 95% confidence interval, we can use the formula: p hat ± 1.96 * SE(p hat). Plugging in the values, we get 0.5375 ± 1.96 * 0.0249, which gives us a confidence interval of (0.4881, 0.5869).

For a 99% confidence interval, we use the same formula but with a different critical value. p hat ± 2.58 * SE(p hat). Plugging in the values, we get 0.5375 ± 2.58 * 0.0249, which gives us a wider confidence interval of (0.4695, 0.6055).