Math  /  Data & Statistics

QuestionAt the final exam in a statistics class, the professor asks each student to indicate how many hours he or she studied for the exam. After grading the exam, the professor computes the least-squares regression line for predicting the final exam score from the number of hours studied. The equation of the line is y=40+2xy=40+2 x.
Part: 0/20 / 2 \square
Part 1 of 2 (a) Antoine studied for 6 hours. What do you predict his exam score will be?
Antoine's exam score is predicted to be 52 . \square
Part: 1/21 / 2 \square
Part 2 of 2 (b) Emma studied for 3 hours longer than Jeremy did. How much higher do you predict Emma's score to be?
Emma's exam score is predicted to be \square points higher than Jeremy's score.

Studdy Solution

STEP 1

1. The least-squares regression line is given by the equation y=40+2x y = 40 + 2x , where y y is the predicted exam score and x x is the number of hours studied.
2. Antoine studied for 6 hours.
3. Emma studied for 3 hours more than Jeremy.

STEP 2

1. Calculate Antoine's predicted exam score.
2. Calculate the difference in predicted scores between Emma and Jeremy.

STEP 3

Substitute x=6 x = 6 into the regression equation to find Antoine's predicted score:
y=40+2(6) y = 40 + 2(6)

STEP 4

Calculate the result:
y=40+12=52 y = 40 + 12 = 52
Antoine's predicted exam score is 52.

STEP 5

Let xj x_j be the number of hours Jeremy studied. Then, Emma studied xj+3 x_j + 3 hours.

STEP 6

Calculate the predicted score for Jeremy:
yj=40+2xj y_j = 40 + 2x_j

STEP 7

Calculate the predicted score for Emma:
ye=40+2(xj+3) y_e = 40 + 2(x_j + 3)

STEP 8

Simplify Emma's predicted score:
ye=40+2xj+6=46+2xj y_e = 40 + 2x_j + 6 = 46 + 2x_j

STEP 9

Find the difference in scores between Emma and Jeremy:
yeyj=(46+2xj)(40+2xj)=6 y_e - y_j = (46 + 2x_j) - (40 + 2x_j) = 6
Emma's exam score is predicted to be 6 points higher than Jeremy's score.
The difference in predicted scores is:
6 \boxed{6}

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