I can see the image contains a homework problem that involves hypothesis testing for two independent random samples, where the population variances are unequal. You are tasked with checking whether there is sufficient evidence to conclude that the two population means ($\mu_1$ and $\mu_2$) are different at the 0.01 significance level.

Here’s how to break this problem down:

Step-by-Step Approach

1. Null and Alternative Hypotheses:

   • The null hypothesis ($H_0$): $\mu_1 = \mu_2$
   • The alternative hypothesis ($H_1$): $\mu_1 \neq \mu_2$

   This indicates a two-tailed test at a significance level of $\alpha = 0.01$.

2. Test Statistic: Since population variances are unequal, you would use Welch’s t-test for independent samples. The formula for the t-statistic in this case is:

   $t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

   Where:

   • $\bar{X}_1$ and $\bar{X}_2$ are the sample means,
   • $s_1^2$ and $s_2^2$ are the sample variances,
   • $n_1$ and $n_2$ are the sample sizes.

3. Degrees of Freedom: The degrees of freedom ($df$) for Welch’s t-test are calculated as:

   $df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}$

4. Conclusion: After calculating the test statistic and degrees of freedom, you would compare the test statistic to the critical value from the t-distribution with the calculated degrees of freedom at $\alpha = 0.01$ for a two-tailed test. If the test statistic exceeds the critical value, you reject the null hypothesis in favor of the alternative, meaning there is significant evidence to conclude that $\mu_1 \neq \mu_2$.

Would you like me to calculate the specific values for this problem using the data provided in the table?

Here are some follow-up questions that build on this:

1. How do you calculate sample variance for each sample?
2. What is the critical value of the t-distribution for $\alpha = 0.01$?
3. How would the approach change if the population variances were equal?
4. What does the p-value represent in the context of hypothesis testing?
5. Can you explain the difference between a one-tailed and two-tailed test?

Tip: Always check whether the population variances are assumed to be equal or unequal when conducting hypothesis tests. This will affect which version of the t-test to use.