Q: Given the regression equation $$\hat{y} = 0.001927 + 0.002387x$$, where $$x is $$SAT score and $$\hat{y} is $$predicted College GPA, what is the predicted GPA for a student with an SAT score of 1200?

$$0.001927 + 0.002387 \times 1200$$

Q: If a student’s SAT score is two standard deviations above the mean, and the correlation $$r = 0.46$$, how many standard deviations above the mean would you predict their College GPA to be?

$$2 \times 0.46 = 0.92$$ standard deviations above the mean

Q: Given the regression equation $$\hat{y}^{1/3} = 0.7383499 + 0.0199410x$$, where $$x is $$the length of a rainbow trout in mm, what is the predicted weight for a trout of length 400 mm?

$$\hat{y} = (0.7383499 + 0.0199410 \times 400)^3$

Question 1

Given the regression equation $$\hat{y} = 0.001927 + 0.002387x$$, where $$x is $$SAT score and $$\hat{y} is $$predicted College GPA, what is the predicted GPA for a student with an SAT score of 1200?

Accepted Answer

$$0.001927 + 0.002387 	imes 1200$$

Question 2

What does the slope $$b_1 = 0.002387$ represent in the context of predicting College GPA from SAT score?

Accepted Answer

For each additional point in SAT score, the predicted College GPA increases by 0.002387 on average.

Question 3

If the correlation between SAT score and College GPA is $$r = 0.46$$, what is the $$R^2$$ value for the regression model?

Accepted Answer

$$R^2$$ = $$(0.46)^2 = 0.2116$

Question 4

Which of the following is NOT an assumption required for linear regression to be appropriate?

Accepted Answer

The $$R^2$$ value must be greater than 0.5.

Question 5

A residual plot shows a clear cone shape, with residuals increasing as $$x$$ increases. What does this indicate?

Accepted Answer

The residuals are heteroscedastic, violating the constant variance assumption.

Question 6

Suppose a student’s SAT score is exactly the average SAT score in the dataset. What would you predict for their College GPA?

Accepted Answer

The predicted GPA would be exactly the average College GPA.

Question 7

If a student’s SAT score is two standard deviations above the mean, and the correlation $$r = 0.46$$, how many standard deviations above the mean would you predict their College GPA to be?

Accepted Answer

$$2 	imes 0.46 = 0.92$$ standard deviations above the mean

Question 8

Which of the following best describes extrapolation in the context of linear regression?

Accepted Answer

Predicting values of $$y$$ for $$x$$ values outside the range of the observed data.

Question 9

Given the regression equation $$\hat{y}^{1/3} = 0.7383499 + 0.0199410x$$, where $$x is $$the length of a rainbow trout in mm, what is the predicted weight for a trout of length 400 mm?

Accepted Answer

$$\hat{y} = (0.7383499 + 0.0199410 	imes 400)^3$

Question 10

Which of the following statements about the intercept $$b_0 in a $$regression equation is correct?

Accepted Answer

It represents the predicted value of $$y$$ when $$x = 0$$, but may not always have a meaningful interpretation.

Question 11

A point in a scatterplot is considered high leverage if:

Accepted Answer

It is extreme in the $$x$$ direction.

Question 12

Which of the following is the formula for the slope $$b_1 of $$the least squares regression line?

Accepted Answer

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

Question 13

Which of the following is the correct formula for the intercept $$b_0 of $$the least squares regression line?

Accepted Answer

$$b_0$$ = \bar{y} - $$b_1$$ \bar{x}$$

Question 14

If the residual plot shows a random scatter around zero, what does this indicate about the linear regression model?

Accepted Answer

The linear regression model is appropriate for the data.

Question 15

Which of the following is true about $$R^2 in $$linear regression?

Accepted Answer

$$R^2 is $$the proportion of the variance in $$y$$ explained by the model.

Question 16

If a point is both high leverage and influential, what does this mean?

Accepted Answer

The point is extreme in $$x$$ and greatly affects the position of the regression line.

Question 17

Which of the following is a valid reason to transform the response variable $$y$$ before fitting a linear regression model?

Accepted Answer

To achieve linearity and/or constant variance in the residuals.

Question 18

Suppose you fit a regression model to predict marathon winning times from year, and the residuals show a clear pattern. What should you conclude?

Accepted Answer

The linear regression model may not be appropriate; consider a different model or transformation.

Question 19

Which of the following best describes the regression effect?

Accepted Answer

Predicted values of $$y$$ are closer to the mean than the corresponding $$x$$ values, unless $$|r| = 1.$$

Linear Regression: Predicting College GPA from SAT Score

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