You are conducting a study to see if the proportion of men over 50 who regularly have their prostate examined is significantly more than 0.87 . You use a significance level of $\alpha=0.05$. \[ \begin{aligned} H_{0}: p & =0.87 \\ H_{1}: p & >0.87 \end{aligned} \] You obtain a sample of size $n=467$ in which there are 413 successes. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic $=923 \sim \sigma^{\circ}$ What is the p-value for this sample? (Report answer accurate to four decimal places.) \[ p \text {-value }=.884 \quad x \] The p-value is... less than (or equal to) $\alpha$ greater than $\alpha$ $\sigma^{\circ}$ This test statistic leads to a decision to... reject the null accept the null fail to reject the null - IMG 2842.jpg $1 M \mathrm{MO} 2841 . j \mathrm{pg}$ IMO 2840 HPg MO 08
Solution
Step 1 ：We are given that the sample size \(n = 467\), the number of successes in the sample is 413, and the hypothesized population proportion \(p_0 = 0.87\).
Step 2 ：First, we calculate the sample proportion \(\hat{p}\) by dividing the number of successes by the sample size: \(\hat{p} = \frac{413}{467} = 0.884\).
Step 3 ：Next, we calculate the test statistic using the formula for a one-sample proportion hypothesis test: \(Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\).
Step 4 ：Substituting the given values into the formula, we get: \(Z = \frac{0.884 - 0.87}{\sqrt{\frac{0.87(1-0.87)}{467}}}\).
Step 5 ：Solving the above expression, we find that the test statistic \(Z = 0.923\).
Step 6 ：\(\boxed{0.923}\) is the test statistic for this sample.
