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. $$ \begin{array}{l}z=-2.58, P \text {-value }=0.0049 \ z=-2.58, P \text {-value }=0.0098 \ z=2.58, P \text {-value }=0.0049 \ z=2.58, P \text {-value }=0.0098\end{array} $$

Question

UpStudy AI · Accepted Answer

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
The test statistic is $$ z = -2.58 $$ and the $$ P $$-value is $$ 0.0098 $$.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Statistics Questions