To solve this hypothesis testing problem, let's go through it step by step.

Step 1: Given Data and Hypotheses

• Significance level ($\alpha$) = 0.01
• Null Hypothesis ($H_0$): $\mu_d = 0$
• Alternative Hypothesis ($H_a$): $\mu_d > 0$
• Sample size ($n$) = 89
• Sample mean difference ($\bar{d}$) = 5.1
• Standard deviation of differences ($s_d$) = 19.9

Step 2: Determine the Critical Value

Since this is a one-tailed test at a 0.01 significance level, we will use the t-distribution to find the critical value. The degrees of freedom (df) for this test are: $df = n - 1 = 89 - 1 = 88$

Using a t-distribution table or calculator for a one-tailed test with $\alpha = 0.01$ and $df = 88$, we find the critical t-value (t$_{0.01, 88}$) to three decimal places.

Let's approximate or calculate this value.

Step 3: Calculate the Test Statistic

The test statistic $t$ is calculated using the formula: $t = \frac{\bar{d}}{s_d / \sqrt{n}}$ Substituting the given values: $t = \frac{5.1}{19.9 / \sqrt{89}}$ Let's compute this result.

Step 4: Decision Rule

• If the calculated test statistic is greater than the critical value, we are in the critical region, and we reject the null hypothesis.
• If the calculated test statistic is less than or equal to the critical value, we are not in the critical region, and we fail to reject the null hypothesis.

Final Calculation and Answer

I’ll perform these calculations for the exact values of the critical value and test statistic.The calculations yield the following results:

• Critical value = 2.369 (rounded to three decimal places)
• Test statistic = 2.418 (rounded to three decimal places)

Conclusion

Since the test statistic (2.418) is greater than the critical value (2.369), the test statistic falls in the critical region.

Decision

This leads us to reject the null hypothesis $H_0$ at the 0.01 significance level.

Would you like more details on any of the steps, or do you have questions?

Here are some related questions to expand on this topic:

1. How would the decision change if the significance level was 0.05 instead of 0.01?
2. How is the test statistic formula derived, and why do we use it in this way?
3. How can we interpret the rejection of the null hypothesis in practical terms for this scenario?
4. What assumptions do we need to validate before conducting this test?
5. How would the results differ if we had a smaller sample size?

Tip: For hypothesis tests, always ensure the assumptions (like normality of differences) hold before concluding.