To determine whether there is a significant difference between the effectiveness of the drug for women and men, we can use a two-sample z-test because the population variances are known. Let's go through the steps for solving this problem.

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Step 1: Define Hypotheses

• Null Hypothesis ($H_0$): There is no difference in effectiveness between women and men ($\mu_w = \mu_m$).
• Alternative Hypothesis ($H_a$): There is a difference in effectiveness between women and men ($\mu_w \neq \mu_m$).

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Step 2: Test Statistic Formula

The formula for the two-sample z-test is:

$z = \frac{\bar{x}_w - \bar{x}_m}{\sqrt{\frac{\sigma^2}{n_w} + \frac{\sigma^2}{n_m}}}$

Where:

• $\bar{x}_w$ and $\bar{x}_m$ are the sample means for women and men.
• $\sigma^2$ is the common population variance.
• $n_w$ and $n_m$ are the sample sizes for women and men.

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Step 3: Plug in Values

Given:

• Women: $n_w = 60$, $\bar{x}_w = 4.4$
• Men: $n_m = 50$, $\bar{x}_m = 3.9$
• Common variance: $\sigma^2 = 3.6$

The test statistic is:

$z = \frac{4.4 - 3.9}{\sqrt{\frac{3.6}{60} + \frac{3.6}{50}}}$

First, calculate the denominator:

$\sqrt{\frac{3.6}{60} + \frac{3.6}{50}} = \sqrt{0.06 + 0.072} = \sqrt{0.132} \approx 0.363$

Now calculate $z$:

$z = \frac{4.4 - 3.9}{0.363} = \frac{0.5}{0.363} \approx 1.377$

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Step 4: Find the p-value

For a two-tailed test with $z = 1.377$, use a standard normal distribution table or a calculator:

$\text{p-value} = 2 \times P(Z > 1.377)$

Using the standard normal table:

$P(Z > 1.377) \approx 0.084$

Thus:

$\text{p-value} = 2 \times 0.084 = 0.168$

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Step 5: Conclusion

Compare the p-value to the significance level ($\alpha = 0.05$):

• Since $0.168 > 0.05$, we fail to reject the null hypothesis.

Interpretation:

There is not enough evidence to conclude that the drug's effectiveness differs between women and men at the 5% significance level.

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Do you want further clarification or have any specific questions? Here are 5 related questions to expand on this:

1. How is the p-value related to the level of significance in hypothesis testing?
2. What would happen if the population variances were unequal?
3. Can this test be conducted as a one-tailed test? If so, how would the conclusion change?
4. What are the assumptions of the two-sample z-test, and are they met in this case?
5. How would increasing the sample size affect the test's sensitivity?

Tip: Always verify whether population variances are equal or known before choosing a z-test or t-test.