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The problem in the image asks to choose the appropriate test to check the hypothesis \( H_0: \mu_1 = \mu_2 \) versus \( H_a: \mu_1 < \mu_2 \) for a given data set, considering that the standard deviations are not the same. The table provides data for two samples with their respective sample means.

### Steps to solve:

1. **Identify the Test:** Since the standard deviations are not equal and the hypothesis involves comparing the means of two independent samples, the appropriate test is **Welch's t-test**. This test is used to compare the means of two independent groups when the assumption of equal variances is violated.

2. **Compute the Test Statistic:** The test statistic for Welch's t-test is calculated as:
   \[
   t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
   \]
   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. **Determine the Degrees of Freedom:** The degrees of freedom for Welch's t-test are calculated using the formula:
   \[
   \text{df} = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}
   \]

4. **Calculate the p-value:** The p-value can be found using a t-distribution with the calculated degrees of freedom.

### Next Steps:
- If you provide the specific values from the data set, I can calculate the exact test statistic and p-value.

Would you like to proceed with these calculations, or do you have any other questions?

### Related Questions:
1. What assumptions are necessary for Welch's t-test?
2. How does Welch's t-test differ from Student's t-test?
3. Can you explain the concept of p-value and how it is interpreted?
4. What are the implications of using a one-tailed test versus a two-tailed test?
5. How do you calculate the confidence interval for the difference in means using Welch's t-test?

### Tip:
Always ensure to check the assumptions of your statistical test, as using the wrong test can lead to incorrect conclusions.

Learn how to choose the appropriate test for comparing the hypothesis H0: μ1 = μ2 versus Ha: μ1 < μ2 using Welch's t-test. This detailed solution covers the steps to compute the test statistic, degrees of freedom, and p-value for two independent samples with unequal variances.

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