3. Determine the critical value, $z_{0}$, to test the claim about the population proportion $p \neq 0.325$ given $n=42$ and $\hat{p}=0.247$ Use $\alpha=0.05$.
Answer
Final Answer: The critical value, \(z_{0}\), for a two-tailed test with \(\alpha = 0.05\) is approximately \(\pm \boxed{1.96}\).
Step 1 ：We are given the sample proportion \(\hat{p} = 0.247\), the population proportion \(p = 0.325\), and the sample size \(n = 42\). We are asked to find the critical value, \(z_{0}\), for a two-tailed test with \(\alpha = 0.05\).
Step 2 ：The formula for calculating the test statistic in a hypothesis test for a population proportion is: \(z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}\).
Step 3 ：Substituting the given values into the formula, we find that the test statistic is approximately -1.08.
Step 4 ：For a two-tailed test with \(\alpha = 0.05\), the critical values are typically around \(\pm 1.96\).
Step 5 ：The test statistic of -1.08 falls within the critical region, so we would reject the null hypothesis that the population proportion is 0.325.
Step 6 ：Finally, we can confirm the critical value using the Z-distribution table. For a two-tailed test with \(\alpha = 0.05\), the critical value is approximately \(\pm 1.96\).
Step 7 ：Final Answer: The critical value, \(z_{0}\), for a two-tailed test with \(\alpha = 0.05\) is approximately \(\pm \boxed{1.96}\).
