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.