Question

question 26 of 39
in professor krugman’s economics course, the correlation between the students’ total scores prior to the final examination and their final - examination scores is r = 0.5. the pre - exam totals for all students in the course have mean 280 and standard deviation 40. the final - exam scores have mean 75 and standard deviation 8. professor krugman has lost julie’s final exam but knows that her total before the exam was 300. he decides to predict her final - exam score from her pre - exam total.
(b) use a regression line to predict julie’s final - exam score. give your answer as a whole number.
y =

Explanation:

Step1: Recall Regression Line Formula

The regression line for predicting \( y \) (final exam score) from \( x \) (pre - exam score) is \( \hat{y}=b_0 + b_1x \), where \( b_1=r\frac{s_y}{s_x} \) and \( b_0=\bar{y}-b_1\bar{x} \).

We are given:

• \( \bar{x} = 280 \) (mean of pre - exam scores), \( s_x=40 \) (standard deviation of pre - exam scores)
• \( \bar{y}=75 \) (mean of final exam scores), \( s_y = 8 \) (standard deviation of final exam scores)
• \( r = 0.5 \) (correlation coefficient)
• \( x = 300 \) (Julie's pre - exam score)

Step2: Calculate the Slope \( b_1 \)

First, calculate the slope \( b_1 \) using the formula \( b_1=r\frac{s_y}{s_x} \).

Substitute the given values: \( r = 0.5 \), \( s_y=8 \), \( s_x = 40 \)

\( b_1=0.5\times\frac{8}{40}=0.5\times0.2 = 0.1 \)

Step3: Calculate the Intercept \( b_0 \)

Use the formula \( b_0=\bar{y}-b_1\bar{x} \)

Substitute \( \bar{y}=75 \), \( b_1 = 0.1 \), \( \bar{x}=280 \)

\( b_0=75-(0.1\times280)=75 - 28=47 \)

Step4: Predict Julie's Final Exam Score

Now, use the regression line \( \hat{y}=b_0 + b_1x \) to predict Julie's final exam score. Substitute \( x = 300 \), \( b_0 = 47 \), \( b_1=0.1 \)

\( \hat{y}=47+0.1\times300=47 + 30=77 \)

Answer:

77