Question

in professor krugmans 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 julies 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.
(a) what is the slope of the least - squares regression line of final - exam scores on pre - exam total scores in this course? give your answer to one decimal place.
slope:
what is the intercept? give your answer as a whole number.
intercept:

Explanation:

Step1: Recall the formula for the slope of the regression line

The formula for the slope \( b \) of the least - squares regression line of \( y \) on \( x \) is \( b = r\times\frac{s_y}{s_x} \), where \( r \) is the correlation coefficient, \( s_y \) is the standard deviation of the response variable (final - exam scores), and \( s_x \) is the standard deviation of the explanatory variable (pre - exam scores).

We are given that \( r = 0.5 \), \( s_y=8 \) (standard deviation of final - exam scores), and \( s_x = 40 \) (standard deviation of pre - exam scores).

Step2: Calculate the slope

Substitute the given values into the formula: \( b=0.5\times\frac{8}{40} \)

First, calculate \( \frac{8}{40}=0.2 \), then multiply by \( 0.5 \): \( 0.5\times0.2 = 0.1 \)

Step3: Recall the formula for the intercept of the regression line

The formula for the intercept \( a \) of the least - squares regression line is \( a=\bar{y}-b\bar{x} \), where \( \bar{y} \) is the mean of the response variable (final - exam scores), \( \bar{x} \) is the mean of the explanatory variable (pre - exam scores), and \( b \) is the slope we just calculated.

We know that \( \bar{y} = 75 \), \( \bar{x}=280 \), and \( b = 0.1 \)

Step4: Calculate the intercept

Substitute the values into the formula: \( a=75-0.1\times280 \)

First, calculate \( 0.1\times280 = 28 \), then subtract from \( 75 \): \( 75 - 28=47 \)

Answer:

Slope: \( 0.1 \)
Intercept: \( 47 \)