To solve this problem, we will follow these steps:

1. Define the null and alternative hypotheses.
2. Calculate the pooled proportion.
3. Calculate the test statistic.
4. Find the p-value.
5. Make a decision based on the p-value and the significance level.

Step 1: Define the Hypotheses

• $H_0: p_1 = p_2$
• $H_a: p_1 \neq p_2$

Step 2: Calculate the Pooled Proportion

The pooled proportion $\hat{p}$ is calculated as follows:

$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$

where:

• $x_1$ = number of successes in the first sample = 399
• $n_1$ = total number in the first sample = 399 + 75 = 474
• $x_2$ = number of successes in the second sample = 204
• $n_2$ = total number in the second sample = 204 + 59 = 263

So,

$\hat{p} = \frac{399 + 204}{474 + 263} = \frac{603}{737} \approx 0.8183$

Step 3: Calculate the Test Statistic

The test statistic for the difference in proportions is calculated using the formula:

$z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p} (1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$

where:

• $\hat{p}_1 = \frac{399}{474} \approx 0.8418$
• $\hat{p}_2 = \frac{204}{263} \approx 0.7753$

So,

$z = \frac{0.8418 - 0.7753}{\sqrt{0.8183 \cdot (1 - 0.8183) \left( \frac{1}{474} + \frac{1}{263} \right)}}$

First, calculate the standard error (SE):

$SE = \sqrt{0.8183 \cdot 0.1817 \left( \frac{1}{474} + \frac{1}{263} \right)} \approx \sqrt{0.8183 \cdot 0.1817 \cdot 0.0078} \approx \sqrt{0.001157} \approx 0.0340$

Now, calculate $z$:

$z = \frac{0.8418 - 0.7753}{0.0340} \approx \frac{0.0665}{0.0340} \approx 1.956$

Step 4: Find the P-Value

Using the standard normal distribution, we find the p-value for $z = 1.956$.

The p-value is twice the area to the right of $|z|$:

$p\text{-value} = 2 \cdot P(Z > 1.956) \approx 2 \cdot 0.0251 = 0.0502$

Step 5: Decision and Conclusion

The significance level $\alpha$ is 0.01.

• Since $p\text{-value} = 0.0502$ is greater than 0.01, we fail to reject the null hypothesis.

Summary of Results

• Test statistic: $z \approx 1.956$
• P-value: $\approx 0.0502$
• The p-value is greater than the significance level (0.01).
• This test statistic leads to a decision to do not reject the null.
• Conclusion: There is not sufficient evidence to warrant rejection of the claim that the first population proportion is not equal to the second population proportion.

Do you want more details or have any questions?

Here are 8 relative questions that expand previous information:

1. How would the results change if the significance level were 0.05 instead of 0.01?
2. What are Type I and Type II errors in the context of this hypothesis test?
3. How can the power of this test be increased?
4. What assumptions must be met for this test to be valid?
5. How does sample size affect the results of a hypothesis test for proportions?
6. What is the interpretation of the p-value in this context?
7. How would you conduct a confidence interval for the difference in proportions?
8. What are alternative methods for comparing two population proportions?

Tip: Always check the assumptions of your statistical test to ensure valid results.