Problem
From previous studies, it is concluded that $24 \%$ of phone numbers in a certain city are unlisted. A researcher claims it has decreased and decides to survey 100 adults. Test the researcher's claim at the $\alpha=0.05$ significance level. Preliminary: ChatGPT 3.5 a. Is it safe to assume that $n \leq 0.05$ of all subjects in the population? Yes No b. Verify $n \hat{p}(1-\widehat{p}) \geq 10$. Round your answer to one decimal place. \[ n \widehat{p}(1-\widehat{p})= \] Test the claim: a. Express the null and alternative hypotheses in symbolic form for this claim. \[ \begin{array}{l} H_{0}: ? \vee ? \sim \\ H_{a}: ? \backsim ? \backsim \end{array} \] b. After surveying 100 adult Americans, the researcher finds that 62 phone numbers in a certain city are unlisted. Compute the test statistic, Round to two decimal places. \[ z= \] c. What is the $p$-value? Round to 4 decimals. \[ p= \]
Solution
Step 1 ：The null hypothesis \(H_{0}: p = 0.24\) and the alternative hypothesis \(H_{a}: p < 0.24\)
Step 2 ：Calculate the test statistic: \(z= \frac{0.62-0.24}{\sqrt{\frac{0.24*(1-0.24)}{100}}} = 7.74\)
Step 3 ：Calculate the p-value: \(p= 0.0000\)
Step 4 ：Since the p-value is less than the significance level of 0.05, we reject the null hypothesis
Step 5 ：This means that the researcher's claim that the proportion of unlisted phone numbers has decreased is supported by the survey data
