Let's break down the tasks in your image.

Part 1:

The research question is: "Is there any difference between paintings and sculptures in the average number of weeks the artworks were exhibited before auction?"

1. Which test is correct to use if we do not know the population standard deviation?

   • Since we don't know the population standard deviation, a two-sample t-test is appropriate, assuming the two groups (paintings and sculptures) have approximately normal distributions and similar variances.

2. What is the null hypothesis?

   • The null hypothesis ($H_0$) would be: There is no difference in the average number of weeks paintings and sculptures were exhibited before the auction (i.e., $\mu_{paintings} = \mu_{sculptures}$).

3. P-value of the test is 0.599. What is the decision of the test?

   • Since the p-value (0.599) is greater than the common significance level (e.g., 0.05), we fail to reject the null hypothesis. This means there is insufficient evidence to suggest a difference in the average exhibition weeks between paintings and sculptures.

Part 2:

The research question is: "Is the average number of weeks the artworks were exhibited before auction equal to 38?"

1. Which test is correct to use if we know the population standard deviation?

   • Since the population standard deviation is known, a z-test for the population mean would be appropriate here.

2. Test statistic is -2.23, and the p-value is 0.027. What is the decision of the test?

   • With a p-value of 0.027, which is less than the significance level (e.g., 0.05), we reject the null hypothesis. This suggests that the average number of weeks the artworks were exhibited is significantly different from 38.

Do you need further details or have any questions?

Here are 5 related questions to expand this topic:

1. How is the assumption of normality tested in a t-test?
2. What are the consequences of not meeting the equal variance assumption in a two-sample t-test?
3. What would the decision be if the p-value were exactly 0.05?
4. How can a confidence interval help in interpreting the results of a t-test?
5. How would the conclusions change if the p-value was higher than 0.05 in Part 2?

Tip: Always check for the assumptions of normality and equal variances before conducting a t-test to ensure valid results.