Let's work through this hypothesis testing problem step-by-step.

Part 1: Determine the Alternative Hypothesis

We are given that the null hypothesis is: $H_0: \mu_1 - \mu_2 \leq 0$ where $\mu_1$ is the mean coverage for the existing paint, and $\mu_2$ is the mean coverage for the new paint.

The developers are concerned that the average area covered per gallon will be less for the new paint than for the existing product. This suggests a one-sided test where they are testing if the new paint has significantly lower coverage than the existing paint.

So, the appropriate alternative hypothesis would be: $H_A: \mu_1 - \mu_2 > 0$ This corresponds to option F.

Part 2: Rejection Region for the Test Statistic $t$

Given that both samples come from normally distributed populations with equal variances, we can conduct a two-sample $t$-test for the difference in means.

1. Compute the Test Statistic $t$:

   The formula for the $t$-statistic in this case is: $t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$ where $s_p$ is the pooled standard deviation, calculated as: $s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$

   Given values:

   • $\bar{x}_1 = 421$
   • $\bar{x}_2 = 407$
   • $s_1 = 23.4$
   • $s_2 = 16.6$
   • $n_1 = 20$
   • $n_2 = 10$

2. Calculate the Degrees of Freedom:

   The degrees of freedom for this $t$-test is: $df = n_1 + n_2 - 2 = 20 + 10 - 2 = 28$

3. Significance Level and Rejection Region:

   At a significance level of $\alpha = 0.05$ for a one-tailed test, we look up the critical $t$-value for 28 degrees of freedom. This value is approximately $t_{0.05, 28} = 1.701$.

   • The rejection region for the test is $t > 1.701$.

Thus, if our calculated $t$-statistic exceeds 1.701, we will reject the null hypothesis in favor of the alternative.

Would you like me to go further and calculate the $t$-statistic for these values, or do you have any questions so far?

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Further Practice Questions:

1. What steps are required to calculate a pooled standard deviation?
2. How would the rejection region change if this were a two-tailed test at the same significance level?
3. How does sample size impact the width of the confidence interval in a two-sample test?
4. What assumptions are necessary for the validity of a two-sample $t$-test?
5. How can effect size be used in addition to statistical significance to interpret results?

Tip: For any hypothesis test, always verify the assumptions for the test before drawing conclusions.