Questions: Assume that in a political science class, the teacher gives a midterm exam and a final exam. Assume that the association between midterm and final scores is linear. The summary statistics shown below have been simplified for clarity. Also, r=0.7 and n=26. Mean Standard deviation ------------------------------------ Midterm 75 10 Final 75 10 According to the regression equation, for a student who gets an 87 on the midterm, what is the predicted final exam grade? What phenomenon does this demonstrate? Explain. The predicted final exam grade is 83 (Round to the nearest integer as needed.) Step 3: Your predicted final exam grade should be less than 87. Why? What phenomenon does this demonstrate? Explain. Choose the correct answer below. A. Regression toward the mean, because the student's predicted score is closer to the mean than was their midterm score. B. Regression toward the mean, because the student's predicted final score is farther from the mean than was their midterm score. C. Extrapolation, because the predicted score is lower than the midterm score. D. Extrapolation, because the score of 87 is outside the range of the data.

Solution Steps

To find the regression line, we first calculate the slope ($\beta$) and intercept ($\alpha$) using the following formulas:

$$\beta = r \cdot \left( \frac{\sigma_Y}{\sigma_X} \right)$$

$$\alpha = \mu_Y - \beta \cdot \mu_X$$

Where:

• $r = 0.7$ (correlation coefficient)
• $\sigma_Y = 10$ (standard deviation of final scores)
• $\sigma_X = 10$ (standard deviation of midterm scores)
• $\mu_Y = 75$ (mean of final scores)
• $\mu_X = 75$ (mean of midterm scores)

Calculating these values gives us:

$$\beta = 0.7 \cdot \left( \frac{10}{10} \right) = 0.7$$

$$\alpha = 75 - 0.7 \cdot 75 = 22.5$$

Using the regression equation, we can predict the final exam grade for a student who scores $87$ on the midterm:

$$\text{Predicted Final Score} = \alpha + \beta \cdot \text{Midterm Score}$$

Substituting the values:

$$\text{Predicted Final Score} = 22.5 + 0.7 \cdot 87$$

Calculating this gives:

$$\text{Predicted Final Score} = 22.5 + 60.9 = 83.4$$

Rounding to the nearest integer, we find:

$$\text{Predicted Final Score} = 83$$

The predicted final exam grade of $83$ is less than the midterm score of $87$. This demonstrates the phenomenon of regression toward the mean. Specifically, since the predicted score is closer to the mean ($75$) than the midterm score, we can conclude:

$$\text{Regression toward the mean, because the student's predicted score is closer to the mean than was their midterm score.}$$

Final Answer