Question 1

Which of the following best describes the purpose of examining residual plots after fitting a regression model?

Accepted Answer

To check for patterns or systematic deviations that indicate the model may not be appropriate.

Question 2

Given the regression equation $$\hat{Y} = 4.72 + 0.70X$$ and an observed value $$Y = 18$$ for $$X = 10$$, what is the residual for this observation?

Accepted Answer

$$7.65$$

Question 3

Which formula provides an estimate for the residual standard deviation $$S_e$$ when given summary statistics?

Accepted Answer

$$S_e \approx S_Y \sqrt{1 - r^2$}$$

Question 4

If the correlation coefficient $$r = 1$$, what can be said about the residual standard deviation $$S_e$$?

Accepted Answer

$$S_e = 0$$

Question 5

A residual plot shows a clear curve (bend) in the residuals. What does this indicate about the regression model?

Accepted Answer

The linear model is not appropriate and a non-linear relationship may exist.

Question 6

Suppose $$S_Y = 5.5$$ and $$r = 0.67. $$Calculate the estimated residual standard deviation $$S_e.$$

Accepted Answer

$$S_e \approx 4.1$$

Question 7

Which of the following is NOT an assumption required for the residual standard deviation formula $$S_e \approx S_Y \sqrt{1 - r^2$} to be $$valid?

Accepted Answer

The correlation coefficient must be negative.

Question 8

In a regression analysis, what does it mean if about 68% of the residuals fall within one $$S_e of $$the regression line?

Accepted Answer

The residuals are approximately normally distributed around the regression line.

Question 9

Given the regression equation $$MWS = 1031.71 - 0.9753 	imes CentralPressure$$ and $$r = -0.898$$, what does the negative slope indicate in the context of hurricanes?

Accepted Answer

As central pressure increases, maximum wind speed decreases.

Question 10

If a residual plot shows increasing spread as X increases, which regression assumption is likely violated?

Accepted Answer

Equal variance assumption (homoscedasticity)

Question 11

A regression model for predicting Math scores from Verbal scores gives $$S_Y = 98.1$$ and $$r = 0.685. $$What is the estimated residual standard deviation $$S_e$$?

Accepted Answer

$$S_e \approx 71.47$$

Question 12

Which of the following statements about residuals is TRUE?

Accepted Answer

Residuals are the differences between observed and predicted values: $$e = Y - \hat{Y}.$$

Question 13

If the residual standard deviation $$S_e is $$close to the standard deviation of Y ($$S_Y$$), what does this suggest about the regression model?

Accepted Answer

The regression model does not explain much of the variation in Y.

Question 14

Which of the following best describes the 'equal variance assumption' in regression analysis?

Accepted Answer

The spread of residuals should be roughly the same for all values of X.

Question 15

A student uses the regression equation $$\hat{Y} = 4.72 + 0.70X to $$predict Exam 2 scores. If $$X = 20$$, what is the predicted Exam 2 score?

Accepted Answer

$$18.72$$

Question 16

Which of the following is a sign that a regression model may be missing important features?

Accepted Answer

The residual plot shows bends or outliers.

Question 17

If $$r = 0$$, what is the value of the estimated residual standard deviation $$S_e$$?

Accepted Answer

$$S_e \approx S_Y$$

Question 18

Which of the following is the correct interpretation of $$S_e in $$the context of regression?

Accepted Answer

$$S_e is $$the average vertical distance of the observed points from the regression line.

Question 19

A regression model predicts Math scores from Verbal scores with $$S_e = 71.755. $$What does this value represent?

Accepted Answer

The typical prediction error (in Math scores) when using the regression model.

Question 20

Which of the following is true about the relationship between $$S_e$$ and $$S_Y$$?

Accepted Answer

$$0 \leq S_e \leq S_Y$$

Examining Residuals and the Residual Standard Deviation in Linear Regression

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