Question

a student at a junior college conducted a survey of 20 randomly selected full - time students to determine the relation between the number of hours of video game playing each week, x, and grade - point average y. she found that a linear relation exists between the two variables. the least - squares regression line that describes this relation is (hat{y}=-0.0502x + 2.9223). assume that observations near (x = 0) exist in the data
(a) predict the grade - point average of a student who plays video games 8 hours per week
the predicted grade - point average is 2.52
(round to the nearest hundredth as needed.)
(b) interpret the slope.
for each additional hour that a student spends playing video games in a week, the grade - point average will
by points, on average

Explanation:

Step1: Substitute \(x = 8\) into the regression line formula

The regression line is \(\hat{y}=-0.0502x + 2.9223\). Substitute \(x = 8\) into it: \(\hat{y}=-0.0502\times8+2.9223\).

Step2: Calculate the value of \(\hat{y}\)

First, calculate \(-0.0502\times8=-0.4016\). Then, \(\hat{y}=-0.4016 + 2.9223=2.5207\approx2.52\).

For part (b), in the linear regression equation \(y = mx + b\) (here \(y=\hat{y}=-0.0502x + 2.9223\)), the slope \(m=-0.0502\). The slope represents the change in \(y\) (grade - point average) for a one - unit change in \(x\) (hours of video - game playing). Since the slope \(m=-0.0502<0\), for each additional hour that a student spends playing video games in a week, the grade - point average will decrease by \(0.0502\) points, on average.

Answer:

(a) \(2.52\)
(b) decrease; \(0.0502\)