Q: Given the regression equation $$\hat{y} = -62.56 + 2.222x$$ for % Body Fat (y) and Waist size (x), what is the predicted % Body Fat for a man with a waist size of 40 inches?

$$\hat{y} = -62.56 + 2.222 \times 40 = 26.32$$

Question 1

Which of the following best describes the relationship between % Body Fat and Waist size in the scatterplot shown in Figure 24.1?

Accepted Answer

There is a strong, positive, linear relationship between % Body Fat and Waist size.

Question 2

In regression analysis, what is the primary purpose of checking a histogram of the residuals, as shown in Figure 24.6?

Accepted Answer

To verify that the residuals are nearly Normal, unimodal, and symmetric.

Question 3

Suppose you fit a regression model and the residuals vs. fits plot shows a clear pattern (e.g., a curve). What does this indicate?

Accepted Answer

The linearity assumption may be violated, and a linear model may not be appropriate.

Question 4

Which of the following is the correct formula for the standard error of the slope in simple linear regression?

Accepted Answer

$$SE(b_1$) = \frac{s_e}{s_x \sqrt{n-1}}$$

Question 5

A 95% confidence interval for the slope $$\$$beta_1$$ in a regression is given by which formula?

Accepted Answer

$$b_1$$ \pm t_{\alpha/2, n-2} \cdot SE($$b_1$$)$$

Question 6

If the residuals from a regression model show increasing spread as the fitted values increase, which assumption is most likely violated?

Accepted Answer

Equal variance (homoscedasticity) assumption.

Question 7

Given the regression equation $$\hat{y} = -62.56 + 2.222x$$ for % Body Fat (y) and Waist size (x), what is the predicted % Body Fat for a man with a waist size of 40 inches?

Accepted Answer

$$\hat{y} = -62.56 + 2.222 	imes 40 = 26.32$$

Question 8

Which of the following best explains why a prediction interval for an individual y-value is wider than a confidence interval for the mean y at a given x?

Accepted Answer

A prediction interval includes both the uncertainty in estimating the mean and the additional variability of individual outcomes.

Question 9

Which of the following is NOT an assumption required for valid inference in simple linear regression?

Accepted Answer

The independent variable x must be normally distributed.

Question 10

You are testing the null hypothesis $$H_0$$: \$$beta_1 = 0$ for the slope in a regression. The output gives b_1$ = 2.222$$, $$SE(b_1$) = 0.273$$, and $$n = 20. $$What is the value of the test statistic?

Accepted Answer

$$t = \frac{2.222}{0.273} = 8.14$$

Question 11

Which of the following best describes the difference between a confidence interval for the mean response and a prediction interval for a new observation at a given value of x?

Accepted Answer

The confidence interval estimates the mean y for all individuals at x, while the prediction interval estimates the y for a single new individual at x.

Question 12

If you observe a pattern in the residuals vs. time plot, what does this suggest about your data?

Accepted Answer

The independence assumption may be violated.

Question 13

Which of the following is the correct formula for a 95% confidence interval for the mean response $$\mu_y at a $$fixed $$x_0$$?

Accepted Answer

$$\hat{y}_0 \pm t_{\alpha/2, n-2} \cdot s_e \sqrt{\frac{1}{n} + \frac{(x_0$ - \bar{x})^2$}{(n-1)s_x^2$}}$$

Question 14

Which of the following is a potential consequence of extrapolating predictions far outside the range of observed x-values in your data?

Accepted Answer

The predictions may be unreliable because the linear relationship may not hold outside the observed range.

Question 15

In the context of regression diagnostics, what does a 'plot thickening' pattern in the residuals vs. fits plot indicate?

Accepted Answer

The variance of the residuals increases with the fitted values, violating the equal variance assumption.

Question 16

Why is it important to check for outliers and high-influence points in regression analysis?

Accepted Answer

Because they can disproportionately affect the slope and intercept, leading to misleading results.

Question 17

Which of the following would most likely result in a more consistent estimate of the regression slope if you were to sample repeatedly from the population?

Accepted Answer

A larger spread in x-values and less scatter around the regression line.

Question 18

If the p-value for the slope in a regression output is much less than $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$; there is significant evidence of a linear relationship between x and y.

Chapter 23: Inferences for Regression – Study Notes

Guida di studio - Domande di pratica

Study guide - Flashcard