Question

A confidence interval for a sample is given, followed by several hypotheses to test using that sample. In each case, use the confidence interval to give a conclusion of the test (if possible) and also state the significance level you are using. A \(95 \%\) confidence interval for \(p: 0.48\) to 0.57 (a) \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (b) \(H_{0}: p=0.75\) vs \(H_{a}: p \neq 0.75\) (c) \(H_{0}: p=0.4\) vs \(H_{a}: p \neq 0.4\)

Short answer

a) Cannot reject \(H_0: p=0.5\) at a 5% significance level. b) Reject \(H_0: p=0.75\) at a 5% significance level. c) Cannot reject \(H_0: p=0.4\) at a 5% significance level.

Step-by-step solution

Understanding the Confidence Interval

Content for Step 1: A 95% confidence interval for the population proportion \(p\) is given as from 0.48 to 0.57. This means that we are 95% confident that the true proportion is within this range.

Test Null Hypothesis (a)

Content for Step 2: The null hypothesis \(H_0: p=0.5\) falls within the range of the confidence interval. Thus, we cannot reject the null hypothesis at a 5% significance level.

Test Null Hypothesis (b)

Content for Step 3: The null hypothesis \(H_0: p=0.75\) falls outside of the range of the confidence interval. Thus, we reject the null hypothesis at a 5% significance level.

Test Null Hypothesis (c)

Content for Step 4: The null hypothesis \(H_0: p=0.4\) falls within the range of the confidence interval. Thus, we cannot reject the null hypothesis at a 5% significance level.