You are conducting a study to see if the proportion of voters who prefer the Democratic candidate is significantly larger than $77 \%$ at a level of significance of $\alpha=0.10$. According to your sample, 58 out of 72 potential voters prefer the Democratic candidate.
a. For this study, we should use z-test for a population proportion
b. The null and alternative hypotheses would be:
Ho:
p
77
(please enter a decimal)
$\infty$
of
o
c. The test statistic $z \neq \gamma=$ (please show your answer to 3 decimal places.)
d. The p-value $=$ (Please show your answer to 4 decimal places.)
e. The p-value is
$\alpha$
f. Based on this, we should fail to reject the null hypothesis.

Step 1 ：We are given that the sample proportion is 58 out of 72, and we want to test if this is significantly larger than 0.77. We can use a one-sample z-test for a population proportion to test this.

Step 2 ：The null hypothesis is that the population proportion is equal to 0.77, and the alternative hypothesis is that the population proportion is greater than 0.77.

Step 3 ：We can calculate the z-score using the formula for the test statistic of a one-sample z-test for a population proportion, which is \((p_{hat} - p_0) / \sqrt{(p_0 * (1 - p_0)) / n}\), where \(p_{hat}\) is the sample proportion, \(p_0\) is the hypothesized population proportion, and n is the sample size.

Step 4 ：After calculating the z-score, we can find the p-value by looking up the z-score in a standard normal distribution table.

Step 5 ：If the p-value is less than the level of significance, we reject the null hypothesis. If the p-value is greater than the level of significance, we fail to reject the null hypothesis.

Step 6 ：Given that n = 72, x = 58, p_0 = 0.77, and alpha = 0.1, we calculate \(p_{hat}\) to be approximately 0.806 and the z-score to be approximately 0.717.

Step 7 ：The p-value is approximately 0.237. Since the p-value is greater than the level of significance (0.10), we fail to reject the null hypothesis.

Step 8 ：This means that we do not have enough evidence to conclude that the proportion of voters who prefer the Democratic candidate is significantly larger than 77%.

Step 9 ：Final Answer: The test statistic \(z\) is approximately \(\boxed{0.717}\) and the p-value is approximately \(\boxed{0.237}\). Therefore, we fail to reject the null hypothesis at a level of significance of 0.10. This means that we do not have enough evidence to conclude that the proportion of voters who prefer the Democratic candidate is significantly larger than 77%.