Let's break down and address each of the questions step by step based on the simple linear regression results provided.

Question 1: What is the null hypothesis?

The null hypothesis ($H_0$) in the context of simple linear regression typically refers to whether there is a relationship between the two variables in the model (i.e., the independent variable and the dependent variable).

Among the options provided, the correct null hypothesis is: d. The slope of the regression line is zero.

Explanation: In simple linear regression, the null hypothesis often tests whether the slope ($\beta_1$) is equal to zero. A slope of zero would imply that there is no relationship between the independent variable (weeks of exhibition) and the dependent variable (price in thousands). If the slope is zero, then changes in the number of exhibition weeks do not affect the sale price.

Question 2: From the test statistic as reported above, we can conclude that the estimated value of the slope is:

The test statistic for the slope of the regression line is given by the t-value, which helps determine whether the slope is significantly different from zero.

• From the regression output, the estimate of the slope is $\hat{\beta_1} = 0.7636$.
• The t-value associated with this estimate is 13.9596, indicating that the slope is highly statistically significant.

Thus, we conclude: The estimated value of the slope is 0.7636.

This means that for each additional week of exhibition, the artwork's price increases by 0.7636 thousand dollars (or $763.60).

Question 3: From the p-value as reported above, we can conclude that the decision of the test is:

The p-value helps us determine whether to reject or fail to reject the null hypothesis.

• From the results, the p-value is 0, which is less than any reasonable significance level (e.g., 0.05, 0.01).
• Since the p-value is 0, we reject the null hypothesis.

Thus, we conclude: Reject the null hypothesis.

Conclusion of the Test:

Since the null hypothesis is rejected, the conclusion is that there is a significant positive relationship between the number of weeks an artwork was exhibited and its sale price. Specifically, the more weeks the artwork was exhibited, the higher its sale price at auction.

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Follow-up Questions:

1. How would the relationship between price and exhibition weeks change if the slope was negative?
2. What assumptions must be satisfied for the linear regression model to be valid?
3. How does the t-value help us determine the significance of the slope in regression?
4. What would the conclusion be if the p-value had been greater than 0.05?
5. How could you interpret the intercept of the model in this context?

Tip:

In regression analysis, always check the p-value to determine the statistical significance of the relationship between variables. A small p-value (typically <0.05) indicates that you can reject the null hypothesis, suggesting a meaningful relationship.