Question 15 You wish to test the following daim $\left(H_{\omega}\right)$ at a significance level of $\alpha=0.005$. \[ \begin{array}{l} H_{e}: p=0.63 \\ H_{e}: p \neq 0.63 \end{array} \] You obtain a sample of size $n=\mathbf{5 2 6}$ in which there are 315 successful observations. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approxamation for the binomial distribution. What is the critical value for this test? (Report answer accurate to three decimal places.) critical value $= \pm$ What is the test statistic for this sample? (Report answer accurate to three decomal places.) test statistic $=$ The test statistic is. in the critical region not in the critical region. This test statistic leads to a decision to. reject the null accept the nul fall to reject the null As such, the final conclusion is that. There is sufficient evidence to warrant rejection of the claim that the population proportion is not equal to 0.63 . There is not sufficient evidence to warrant rejection of the claim that the population proportion is not equal to 0.63 . The sample data support the dam that the population proporton is not equal to 0.63 . There is not sufficient sample evidence to support the claim that the population proportion is not equal to 0.63. Check Answer
Solution
Step 1 ：Given values are: significance level \(\alpha = 0.005\), population proportion \(p = 0.63\), sample size \(n = 526\), and number of successful observations \(x = 315\).
Step 2 ：Calculate the sample proportion \(\hat{p} = \frac{x}{n} = \frac{315}{526} = 0.5989\).
Step 3 ：Calculate the critical value using the normal distribution approximation for the binomial distribution. The critical value is the z-score that corresponds to the given significance level \(\alpha\). The critical value is \(z_{\alpha/2} = 2.807\).
Step 4 ：Calculate the standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.63(1-0.63)}{526}} = 0.0211\).
Step 5 ：Calculate the test statistic using the formula \(z = \frac{\hat{p} - p}{\sigma} = \frac{0.5989 - 0.63}{0.0211} = -1.479\).
Step 6 ：Since the test statistic is less than the critical value, it is not in the critical region. Therefore, we fail to reject the null hypothesis.
Step 7 ：As such, the final conclusion is that there is not sufficient evidence to warrant rejection of the claim that the population proportion is not equal to 0.63.
Step 8 ：Final Answer: Critical value = \(\boxed{2.807}\), Test statistic = \(\boxed{-1.479}\). The test statistic is not in the critical region. This test statistic leads to a decision to fail to reject the null. As such, the final conclusion is that there is not sufficient evidence to warrant rejection of the claim that the population proportion is not equal to 0.63.
