Q: Given the regression equation $$\hat{y} = 100 - 5.27x$$, what is the predicted final grade for a student who skipped 4 classes?

$$100 - 5.27 \times 4 = 78.92$$

Question 1

Which of the following best describes the slope in a simple linear regression model?

Accepted Answer

It indicates the expected change in $$y$$ for a one-unit increase in $$x.$$

Question 2

Given the regression equation $$\hat{y} = 100 - 5.27x$$, what is the predicted final grade for a student who skipped 4 classes?

Accepted Answer

$$100 - 5.27 	imes 4 = 78.92$$

Question 3

What does the intercept $$b_0 in $$the regression equation $$\hat{y} = b_0$ + b_1x$$ represent?

Accepted Answer

The predicted value of $$y$$ when $$x = 0$$

Question 4

If the correlation coefficient $$r$$ between $$x$$ and $$y is $$negative, what can you say about the slope $$b_1 of $$the regression line?

Accepted Answer

The slope $$b_1$$ will be negative.

Question 5

Which formula correctly calculates the slope $$b_1 of $$the least squares regression line?

Accepted Answer

$$b_1 = r$$ \frac{s_y}{s_x}$$

Question 6

A car dealership uses the regression equation $$	ext{price} = 19583 - 1042 	imes 	ext{age} to $$predict car prices. What is a potential problem with using this model to predict the price of a 20-year-old car?

Accepted Answer

Extrapolation: The model may not be valid outside the observed age range.

Question 7

What is the definition of a residual in linear regression?

Accepted Answer

The difference between the observed value $$y$$ and the predicted value $$\hat{y}.$$

Question 8

Why is it important to examine the residual plot after fitting a linear regression model?

Accepted Answer

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

Question 9

If the sum of the residuals from a regression model is not zero, what does this indicate?

Accepted Answer

There is likely a calculation error, as the sum should always be zero.

Question 10

Suppose the correlation coefficient between two variables is $$r = 0.7. $$What proportion of the variation in $$y is $$explained by the regression line?

Accepted Answer

$$r^2 = 0.49$, so 49\%$

Question 11

Which of the following is a correct interpretation of the regression line $$\hat{y} = 100 - 5.27x in $$the context of predicting final grades based on classes skipped?

Accepted Answer

For each additional class skipped, the predicted final grade decreases by 5.27 points.

Question 12

Which of the following statements about extrapolation in linear regression is TRUE?

Accepted Answer

Extrapolation can lead to unreliable predictions because the linear relationship may not hold outside the observed data range.

Question 13

If the standard deviation of the residuals $$s_e is $$close to the standard deviation of the original $$y$$ data, 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 equation gives the variance of the residuals in terms of the variance of $$y$$ and the correlation coefficient $$r$$?

Accepted Answer

$$s_e^2$ = (1 - r^2$) 	imes 	ext{Var}(y)$$

Question 15

A residual plot shows a clear pattern (e.g., a curve or funnel shape). What does this indicate about the linear regression model?

Accepted Answer

The linear model may not be appropriate for the data.

Question 16

Which of the following is a lurking variable in the context of predicting final grades based on classes skipped?

Accepted Answer

Student motivation, which may affect both attendance and grades.

Question 17

If a student skipped 30 classes, is it reasonable to use the regression line $$\hat{y} = 100 - 5.27x to $$predict their final grade?

Accepted Answer

No, because this is extrapolation beyond the observed data range.

Question 18

Which of the following is NOT a property of the least squares regression line?

Accepted Answer

It always passes through every data point.

Question 19

In the regression equation $$\hat{y} = b_0$ + b_1x$$, what does $$\hat{y}$$ represent?

Accepted Answer

The predicted value of $$y$$ for a given $$x$$

Linear Regression: Concepts, Computation, and Interpretation

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