Problem
Week 7 Homework Score: \( 24.58 / 30 \quad 29 / 30 \) answered Question 26 Hypothesis Test for a Population Proportion A well-known brokerage firm executive claimed that \( 80 \% \) of investors are currently confident their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, a sample of 200 people, \( 74 \% \) of them said they are confident of meeting their goals. Test the claim that the proportiof of people who are confident is smaller than \( 80 \% \) at the 0.01 level. The null and alternative hypothesis would be: \[ \begin{array}{l} H_{0}: \mu=0.8 H_{0}: \mu \geq 0.8 \quad H_{0}: p \leq 0.8 \quad H_{0}: p=0.8 \quad H_{0}: \mu \leq 0.8 \quad H_{0}: p \geq 0.8 \\ H_{1}: \mu \neq 0.8 H_{1}: \mu<0.8 H_{1}: p>0.8 H_{1}: p \neq 0.8 \quad H_{1}: \mu>0.8 \quad H_{1}: p<0.8 \\ \end{array} \] The test is: left-tailed two-tailed right-tailed Pype here to search
Solution
Step 1 ：Define the null hypothesis \(H_0\) and alternative hypothesis \(H_1\): \(H_0: p = 0.8\), \(H_1: p < 0.8\)
Step 2 ：Calculate the test statistic: \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} = \frac{0.74 - 0.8}{\sqrt{\frac{0.8(1 - 0.8)}{200}}}\)
Step 3 ：Find the p-value for the obtained test statistic, and compare it to the significance level \(0.01\)
