Let's address each part of the problem step-by-step.

Part (a)

1. Identify the hypotheses: Given the question states to test if $\mu_1 > \mu_2$, the hypotheses are:

   • Null Hypothesis ($H_0$): $\mu_1 = \mu_2$
   • Alternative Hypothesis ($H_1$): $\mu_1 > \mu_2$

   The correct answer for Part (a) is therefore Option D:

   • $H_0: \mu_1 = \mu_2$
   • $H_1: \mu_1 > \mu_2$

2. Test Statistic: You’ve calculated the test statistic as $1.06$.

3. P-value: The P-value provided is $0.147$.

4. Conclusion: Since the P-value $0.147$ is greater than the significance level $\alpha = 0.05$, we do not reject the null hypothesis.

   Therefore, the correct conclusion is Option A:

   • Do not reject $H_0$.
   • There is not sufficient evidence at the $\alpha = 0.05$ level of significance to conclude that $\mu_1 > \mu_2$.

Part (b)

For the 90% confidence interval for $\mu_1 - \mu_2$, we'll use the following formula:

$\text{Confidence Interval} = \left(\bar{x}_1 - \bar{x}_2\right) \pm t_{\alpha/2} \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

where:

• $\bar{x}_1 = 45.1$, $s_1 = 6.6$, $n_1 = 28$
• $\bar{x}_2 = 42.4$, $s_2 = 10.2$, $n_2 = 21$
• $t_{\alpha/2}$ corresponds to the critical value for a 90% confidence level with degrees of freedom calculated using the approximation formula.

Let’s calculate each step:

1. Difference in sample means: $\bar{x}_1 - \bar{x}_2 = 45.1 - 42.4 = 2.7$

2. Standard error:

   $SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{6.6^2}{28} + \frac{10.2^2}{21}}$

3. Degrees of freedom: Approximate degrees of freedom can be calculated, and we can then find the corresponding $t$-value for the 90% confidence level.

After calculation, substitute the values to obtain the confidence interval bounds. I’ll calculate this to provide the precise lower and upper bounds for the confidence interval.

The 90% confidence interval for $\mu_1 - \mu_2$ is approximately:

$(-1.621, 7.021)$

This interval suggests that the difference between $\mu_1$ and $\mu_2$ could range from about -1.621 to 7.021. Since the interval includes zero, it aligns with the conclusion from the hypothesis test in Part (a) that there isn’t sufficient evidence to claim $\mu_1 > \mu_2$ at the 0.05 significance level.

Would you like further clarification on any part of this solution, or do you have any additional questions?

Here are five follow-up questions that could deepen your understanding:

1. What would change if we used a 95% confidence level instead of a 90% level for the confidence interval?
2. How does the standard deviation impact the width of the confidence interval?
3. Why does the confidence interval contain zero, and what does this imply about the populations?
4. How would the results differ if the sample sizes of Population 1 and Population 2 were larger?
5. What are the assumptions made for conducting this hypothesis test and confidence interval?

Tip: In hypothesis testing, always check the P-value against your significance level before drawing conclusions about the null hypothesis.