A store manager claims that \( 60 \% \) of shoppers who enter her store make a purchase. To investigate this claim, she selects a random sample of 40 customers and finds that \( 40 \% \) of them make a purchase. She wants to know if the data provide convincing evidence that the true proportion of all customers entering her store who make a purchase differs from \( 60 \% \). What are the values of the test statistic and \( P \)-value for this test? Find the \( z \)-table here. \[ \begin{array}{l}z=-2.58, P \text {-value }=0.0049 \\ z=-2.58, P \text {-value }=0.0098 \\ z=2.58, P \text {-value }=0.0049 \\ z=2.58, P \text {-value }=0.0098\end{array} \]

To determine whether the store manager's claim that \( 60\% \) of shoppers make a purchase is supported by the data, we'll perform a hypothesis test for a population proportion. ### Step 1: Define the Hypotheses - **Null Hypothesis (\( H_0 \))**: The true proportion of customers making a purchase is \( 60\% \). \( H_0: p = 0.60 \) - **Alternative Hypothesis (\( H_a \))**: The true proportion of customers making a purchase is not \( 60\% \). \( H_a: p \neq 0.60 \) ### Step 2: Calculate the Test Statistic The test statistic for a proportion is calculated using the formula: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] where: - \( \hat{p} = 0.40 \) (sample proportion) - \( p_0 = 0.60 \) (hypothesized proportion) - \( n = 40 \) (sample size) Plugging in the values: \[ \sqrt{\frac{0.60 \times 0.40}{40}} = \sqrt{\frac{0.24}{40}} = \sqrt{0.006} \approx 0.07746 \] \[ z = \frac{0.40 - 0.60}{0.07746} \approx \frac{-0.20}{0.07746} \approx -2.58 \] ### Step 3: Determine the \( P \)-value Since this is a two-tailed test, we need to find the probability that \( Z \) is less than \( -2.58 \) or greater than \( 2.58 \). Using the \( z \)-table: - The area to the left of \( z = -2.58 \) is approximately \( 0.0049 \). - Since it's two-tailed, multiply by 2: \( P\text{-value} = 2 \times 0.0049 = 0.0098 \). ### Conclusion The test statistic is \( z = -2.58 \) and the \( P \)-value is \( 0.0098 \). **Answer:** z = –2.58 and the P-value is 0.0098