MATH1127: Introduction to Statistics (60022) Lesson 9.3 Distribution Needed for Hypothesis Testing Unit 3 Chapter 9: Lesso Unit 3 Chapter 9: Lesson 9.3 Assignment Score: $28 / 100$ $4 / 14$ answered Question 13 A well-known brokerage firm executive claimed that at least $64 \%$ of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, found that out of 67 randomly selected people, 46 of them said they are confident of meeting their goals. Suppose you are have the following null and alternative hypotheses for a test you are running: \[ \begin{array}{l} H_{0}: p=0.64 \\ H_{a}: p>0.64 \end{array} \] Calculate the test statistic, rounded to 3 decimal places \[ z= \] Question Help: D Post to forum Submit Question
Solution
Step 1 ：The problem provides the following hypotheses for a test: \(H_{0}: p=0.64\) and \(H_{a}: p>0.64\). It also provides the sample size (n=67) and the number of positive responses (46).
Step 2 ：We first calculate the sample proportion (\(\hat{p}\)) which is the ratio of the number of positive responses to the sample size. So, \(\hat{p} = \frac{46}{67} = 0.6865671641791045\).
Step 3 ：The hypothesized population proportion (\(p_0\)) is given as 0.64.
Step 4 ：We can now calculate the test statistic (z) using the formula: \[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]
Step 5 ：Substituting the values into the formula, we get: \[z = \frac{0.6865671641791045 - 0.64}{\sqrt{\frac{0.64(1-0.64)}{67}}} = 0.7941013883159834\]
Step 6 ：Rounding to 3 decimal places, the test statistic is \(\boxed{0.794}\).
