A publisher reports that 41%41% of their readers own a particular make of car.  A marketing executive wants to test the claim that the percentage is actually different from the reported percentage.  A random sample of 240240 found that 35%35% of the readers owned a particular make of car.  Determine the P-value of the test statistic.  Round your answer to four decimal places.

We are tasked with determining the **P-value** of the test statistic for the marketing executive's hypothesis test. The null hypothesis (\( H_0 \)) is that the proportion of readers who own the car is the same as reported, i.e., \( p = 0.41 \), and the alternative hypothesis (\( H_a \)) is that the true proportion is different, i.e., \( p \neq 0.41 \).

### Step 1: Set up the null and alternative hypotheses
- \( H_0: p = 0.41 \)
- \( H_a: p \neq 0.41 \) (two-tailed test)

### Step 2: Calculate the test statistic
The test statistic for a proportion is given by the formula:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}
\]
Where:
- \( \hat{p} \) = sample proportion = 0.35
- \( p_0 \) = population proportion under the null hypothesis = 0.41
- \( n \) = sample size = 240

### Step 3: Plug in the values
\[
z = \frac{0.35 - 0.41}{\sqrt{\frac{0.41(1 - 0.41)}{240}}}
\]
Now, let's compute this.The test statistic \( z \) is approximately \(-1.8899\).

### Step 4: Determine the P-value
Since this is a two-tailed test, the P-value is calculated by finding the area in both tails of the standard normal distribution corresponding to this test statistic.

To find the P-value, we calculate the probability that \( Z \) is less than \(-1.8899\) (and double it, as it's a two-tailed test). Let's compute that next.The P-value is approximately \( 0.0588 \) (rounded to four decimal places).

### Conclusion:
Since the P-value is \( 0.0588 \), it suggests moderate evidence against the null hypothesis at typical significance levels like \( \alpha = 0.05 \). However, the result is not significant at \( \alpha = 0.05 \), but it would be significant at \( \alpha = 0.10 \).

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How is the decision to reject or fail to reject the null hypothesis made using the P-value?
2. What would happen to the P-value if the sample size were larger?
3. How does a two-tailed test differ from a one-tailed test?
4. What are the consequences of using a significance level \( \alpha = 0.10 \) instead of \( \alpha = 0.05 \)?
5. How does changing the population proportion \( p_0 \) affect the test statistic?

**Tip:** Always define your hypotheses clearly before conducting any hypothesis test to ensure correct interpretation of the results.

A publisher reports that 41% of their readers own a particular make of car. A marketing executive wants to test the claim that the percentage is actually different from the reported percentage. A random sample of 240 found that 35% of the readers owned a particular make of car. Determine the P-value of the test statistic. Round your answer to four decimal places.

Determine the P-value of the test statistic for a two-tailed hypothesis test. This example involves testing whether 35% of a sample differs from the claimed 41% car ownership rate using hypothesis testing, proportions, and the z-test formula. Suitable for advanced high school and college-level statistics students.

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