b) A poll showed the approval rating to be $0.52(52 \%)$. A second poll based on 1500 randomly selected voters showed that 804 approved of the job the president was doing. Do the results of the second poll indicate that the proportion of voters who approve of the job the president is doing is significantly higher than the original level? Explain. Assume the $\alpha=0.05$ level of significance.
Identify the null and alternative hypotheses for this test.
A.
\[
\begin{array}{l}
H_{0}: p \neq 0.52 \\
H_{1}: p=0.52
\end{array}
\]
B. $\mathrm{H}_{0}: \mathrm{p}=\mathrm{a} 52$
\[
H_{1}: p=0.52
\]
c.
\[
\begin{array}{l}
H_{0}: p=0.52 \\
H_{1}: p \neq 0.52 \\
H_{0}: p=0.52 \\
H_{1}: p> 0.52
\end{array}
\]
D. $H_{0}: p< 0.52$
E. $H_{0}: p=0.52$
$H_{1}: p=0.52$
$\mathrm{H}_{1}: \mathrm{p}< 0.52$
$H_{0}: p=0.52$
$H_{1}: p> 0.52$
Find the test statistic for this hypothesis test.
$z=\square$ (Round to two decimal places as needed.)
Answer
Find the test statistic for this hypothesis test. The test statistic is a measure of how far our sample statistic is from the null hypothesis value, in standard deviation units. For a proportion, the test statistic is calculated as: \[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\] where \(\hat{p}\) is the sample proportion, \(p_0\) is the null hypothesis value, and \(n\) is the sample size. In this case, \(\hat{p} = \frac{804}{1500}\), \(p_0 = 0.52\), and \(n = 1500\). The test statistic for this hypothesis test is \(\boxed{1.24}\).
Step 1 ：Identify the null and alternative hypotheses for this test. The null hypothesis is a statement of no effect or no difference and is assumed to be true until we have evidence to the contrary. The alternative hypothesis is a statement of an effect or difference and is what we are trying to provide evidence for. In this case, we are testing whether the proportion of voters who approve of the job the president is doing is significantly higher than the original level of 0.52. Therefore, the null hypothesis should be that the proportion is equal to 0.52 and the alternative hypothesis should be that the proportion is greater than 0.52. The correct hypotheses are: \[H_{0}: p=0.52\] \[H_{1}: p>0.52\]
Step 2 ：Find the test statistic for this hypothesis test. The test statistic is a measure of how far our sample statistic is from the null hypothesis value, in standard deviation units. For a proportion, the test statistic is calculated as: \[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\] where \(\hat{p}\) is the sample proportion, \(p_0\) is the null hypothesis value, and \(n\) is the sample size. In this case, \(\hat{p} = \frac{804}{1500}\), \(p_0 = 0.52\), and \(n = 1500\). The test statistic for this hypothesis test is \(\boxed{1.24}\).
