Question

Approval Rating for Congress In a Gallup poll \(^{51}\) conducted in December 2015 , a random sample of \(n=824\) American adults were asked "Do you approve or disapprove of the way Congress is handling its job?" The proportion who said they approve is \(\hat{p}=0.13,\) and a \(95 \%\) confidence interval for Congressional job approval is 0.107 to 0.153 . If we use a 5\% significance level, what is the conclusion if we are: (a) Testing to see if there is evidence that the job approval rating is different than \(14 \%\). (This happens to be the average sample approval rating from the six months prior to this poll.) (b) Testing to see if there is evidence that the job approval rating is different than \(9 \%\). (This happens to be the lowest sample Congressional approval rating Gallup ever recorded through the time of the poll.)

Short answer

(a) There is not enough evidence to conclude that the approval rating is different than 14%. (b) There is enough evidence to conclude that the approval rating is different than 9%.

Step-by-step solution

State the Null and Alternative Hypotheses

(a) Here we are testing to see if there is evidence that the job approval rating is different than 14%. Thus, the null hypothesis (\(H_0\)) is that the approval rating (\(p\)) equals 14%, \(H_0: p = 0.14\), and the alternative hypothesis (\(H_a\)) is that the approval rating (\(p\)) is different from 14%, \(H_a: p \neq 0.14\).\n\n(b) Similarly, here we are testing if the approval rating is different than 9%. So, the null hypothesis (\(H_0\)) is that the approval rating equals 9%, \(H_0: p = 0.09\), and the alternative hypothesis is that the approval rating is different than 9%, \(H_a: p \neq 0.09\).

Interpret the Confidence Interval

A 95% confidence interval for approval is from 0.107 to 0.153. This tells us that based on this sample, we are 95% confident that the true approval rating lies within this interval.

Conclusions of the Tests

(a) Because 14% is within our confidence interval (0.107 to 0.153), we do not reject the null hypothesis and conclude that there is not enough evidence to say that the approval rating is different than 14%.\n\n(b) Because 9% is not within our confidence interval (0.107 to 0.153), we reject the null hypothesis and conclude that there is enough evidence to say that the approval rating is different than 9%.