Question

According to a Washington Post- \(A B C\) News poll, 331 of 502 randomly selected U.S. adults interviewed said they would not be bothered if the National Security Agency collected records of personal telephone calls they had made. Is there sufficient evidence to conclude that a majority of U.S. adults feel this way? Test the appropriate hypotheses using a \(.01\) significance level.

Short answer

Based on the hypothesis test at a 0.01 significance level, there is sufficient evidence to conclude that the majority of U.S. adults wouldn't be bothered if the NSA collected records of personal phone calls.

Step-by-step solution

Understand the Question

The aim is to determine if there is enough evidence to conclude that a majority of U.S. adults would not be bothered if the NSA collected records of personal phone calls they've made. This is a question of hypothesis testing about a population proportion. The null hypothesis (\(H_0\)) is that the proportion of adults not bothered (p) is 0.50 (50%) and the alternate hypothesis (\(H_1\)) is that the proportion of adults not bothered (p) is greater than 0.50 (50%). We're given that our significance level (\(\alpha\)) is 0.01.

Calculate Test Statistic

We first calculate our sample proportion (\(\hat{p}\)) which equals the number of successes (adults not bothered) divided by the sample size. So, \(\hat{p} = 331/502 = 0.6594\). The next task is to calculate the test statistic which equals \(Z = (\hat{p} - p_0)/\sqrt{(p_0 * (1 - p_0))/n}\) where \(p_0\) is the assumed population proportion under the null hypothesis and n is the sample size. Here, \(Z = (0.6594 - 0.50)/ \sqrt{(0.50 * (1 - 0.50))/502} = 9.4846\).

Reject or Fail to Reject Null Hypothesis

Using a standard Z-table or a Z-score calculator, we find that the p-value associated with our test statistic (Z = 9.4846) is less than our significance level of .01. Therefore, we reject the null hypothesis. This suggests that there is sufficient evidence to support the claim that a majority of U.S. adults feel the same way