To solve this problem, we will follow these steps:

Part 1: Hypotheses

Given that we need to test whether $\mu_1 - \mu_2 \neq 0$ at $\alpha = 0.02$, the correct hypotheses are:

• Null Hypothesis (H₀): $\mu_1 - \mu_2 = 0$ (There is no difference between the population means)
• Alternative Hypothesis (Hₐ): $\mu_1 - \mu_2 \neq 0$ (There is a significant difference between the population means)

Therefore, the correct choice is:

$\boxed{B. \ H_0: \mu_1 - \mu_2 = 0 \ , \ H_a: \mu_1 - \mu_2 \neq 0}$

Part 2: Test Statistic

We are performing a two-sample $t$-test for unequal variances (also known as Welch’s t-test). To compute the test statistic $t$, we use the formula:

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

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.

Let's calculate the necessary values:

Step 1: Calculate the sample means:

For Sample 1: $49, 32, 46, 53, 55, 44, 33, 53, 33, 44$

$\bar{X}_1 = \frac{49 + 32 + 46 + 53 + 55 + 44 + 33 + 53 + 33 + 44}{10} = \frac{442}{10} = 44.2$

For Sample 2: $30, 34, 64, 60, 50, 22, 50, 24, 34, 64$

$\bar{X}_2 = \frac{30 + 34 + 64 + 60 + 50 + 22 + 50 + 24 + 34 + 64}{10} = \frac{432}{10} = 43.2$

Step 2: Calculate the sample variances:

For Sample 1: $s_1^2 = \frac{\sum{(X_1 - \bar{X}_1)^2}}{n_1 - 1}$

For Sample 2: $s_2^2 = \frac{\sum{(X_2 - \bar{X}_2)^2}}{n_2 - 1}$

Let me calculate the variances and then the test statistic.It seems like I can’t do more advanced data analysis right now. Please try again later. If you want, you can try calculating the variances and test statistic on your own based on the steps I provided.

If you need further help, feel free to ask! Meanwhile, here are some related questions to deepen your understanding:

1. What assumptions are needed for Welch’s t-test?
2. How do you interpret the p-value in hypothesis testing?
3. What’s the difference between a one-tailed and two-tailed test?
4. How does the unequal variance assumption affect the t-test?
5. Why do we use sample variances instead of population variances in hypothesis tests?

Tip: Always check the assumptions of normality and variance before selecting the appropriate test method.