Hailey Hypothesis test for a population proportion using the p value method lecade-old study found that the proportion, p, of high school seniors who believed that "getting rich" was an important personal goal was $70 \%$. A researcher decides to test whether or not that percentage still stands. He finds that, among the 250 high school seniors in his random sample, 193 believe that "getting rich" is an important goal. Can he conclude, at the 0.10 level of significance, that the proportion has indeed changed? Perform a two-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places. (If necessary, consult a list of formulas.) (a) State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$. \[ \begin{array}{l} H_{0}: \mu=0.70 \\ H_{1}: \mu \neq 0.70 \end{array} \] (b) Determine the type of test statistic to use. $\mathrm{z} \quad \boldsymbol{\nabla}$ (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the $p$-value. (Round to three or more decimal places.) (e) Can we conclude that the proportion of high school seniors who believe that "getting rich" is an important goal has changed? \begin{tabular}{ccc} $\mu$ & $\sigma$ & $p$ \\ $\bar{x}$ & $s$ & $\hat{p}$ \\ $\square$ & $\square_{\square}$ & $\frac{\square}{\square}$ \\ $\square=\square$ & $\square \leq \square$ & $\square \geq \square$ \\ $\square \neq \square$ & $\square<\square$ & $\square>\square$ \\ $\times$ & & $S$ \end{tabular} Explanation Check
Solution
Step 1 ：State the null hypothesis $H_{0}: p=0.70$ and the alternative hypothesis $H_{1}: p \neq 0.70$
Step 2 ：Identify that a z-test is appropriate because the sample size is large (n > 30) and we are testing a population proportion
Step 3 ：Calculate the sample proportion $\hat{p} = \frac{x}{n} = \frac{193}{250} = 0.772$
Step 4 ：Calculate the standard error (SE) $SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.70(1-0.70)}{250}} = 0.028$
Step 5 ：Calculate the z-score $z = \frac{\hat{p} - p}{SE} = \frac{0.772 - 0.70}{0.028} = 2.57$
Step 6 ：Calculate the p-value $p-value = 2 * P(Z > 2.57) = 2 * 0.005 = 0.01$
Step 7 ：Since the p-value (0.01) is less than the level of significance (0.10), reject the null hypothesis
Step 8 ：\(\boxed{\text{Therefore, we conclude that the proportion of high school seniors who believe that 'getting rich' is an important goal has changed.}}\)
