Linear Regression: Predicting College GPA from SAT Score

Q1. Which GPA would the model predict for a student with a SAT score of 1000?

Background

Topic: Linear Regression Prediction

This question tests your ability to use the linear regression equation to predict the response variable (College GPA) from the explanatory variable (SAT Score).

Key Terms and Formulas

• Regression Equation: $\hat{y} = b_0 + b_1 x$

• $b_0$ = Intercept (here, 0.001927)

• $b_1$ = Slope (here, 0.002387)

• $x$ = SAT Score (here, 1000)

Step-by-Step Guidance

1. Identify the regression equation from the model output: $\hat{y} = 0.001927 + 0.002387 x$

2. Plug in the SAT score value ($x = 1000$) into the equation.

3. Set up the calculation: $\hat{y} = 0.001927 + 0.002387 \times 1000$

4. Carefully check the order of operations and units to ensure the calculation is correct.

Try solving on your own before revealing the answer!

Final Answer: $\hat{y} = 0.001927 + 0.002387 \times 1000$

Plugging in the values, the predicted GPA for a student with a SAT score of 1000 is calculated using the regression equation above.

This setup matches answer choice A in the multiple-choice options.

Q2. What would you predict for the GPA of a student whose SAT score is exactly the average?

Background

Topic: Regression Line Properties

This question tests your understanding of the regression line's behavior at the mean of the explanatory variable.

Key Terms and Formulas

• Regression line always passes through $(\bar{x}, \bar{y})$ (mean SAT score, mean GPA).

• If $x = \bar{x}$, then $\hat{y} = \bar{y}$.

Step-by-Step Guidance

1. Recall that the regression line passes through the point $(\bar{x}, \bar{y})$.

2. When the SAT score is exactly the average ($x = \bar{x}$), substitute this value into the regression equation.

3. Observe that $\hat{y} = b_0 + b_1 \bar{x}$, which simplifies to $\bar{y}$.

Try solving on your own before revealing the answer!

Final Answer: The predicted GPA score would be exactly the average.

This is a fundamental property of the least squares regression line.

Q3. The correlation between GPA and SAT score is $r = 0.46$. What would you predict for the GPA of a student, if their SAT score was two standard deviations above the mean?

Background

Topic: Standardized Prediction in Linear Regression

This question tests your understanding of how the regression line relates standardized values of the explanatory and response variables.

Key Terms and Formulas

• Standardized values: $z_x = \frac{x - \bar{x}}{s_x}$, $z_y = \frac{y - \bar{y}}{s_y}$

• Regression effect: If $x$ is $k$ standard deviations above the mean, predicted $y$ is $r \times k$ standard deviations above the mean.

• Here, $k = 2$, $r = 0.46$

Step-by-Step Guidance

1. Identify the number of standard deviations above the mean for SAT score: $k = 2$.

2. Recall the regression effect: predicted GPA is $r \times k$ standard deviations above the mean.

3. Set up the calculation: $r \times 2 = 0.46 \times 2$

4. Interpret the result: The predicted GPA is $0.92$ standard deviations above the mean.

Try solving on your own before revealing the answer!

Final Answer: We would predict the student's GPA to be $0.92$ standard deviations above the mean.

This follows directly from the regression effect formula: $r \times k$.