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 } = - 0.0502 x + 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 decrease by 0.0502 points, on average
(c) if appropriate, interpret the y - intercept
a. the average number of video games played in a week by students is 2.9223
b. the grade - point average of a student who does not play video games is 2.9223
c. it cannot be interpreted without more information

Explanation:

Step1: Substitute \(x = 0\) into the regression equation

The regression equation is \(\hat{y}=-0.0502x + 2.9223\). When \(x = 0\) (since \(x\) represents hours of video - game playing and we want to find the \(y\) - intercept which is the value of \(y\) when \(x = 0\)), we substitute \(x = 0\) into the equation:
\(\hat{y}=-0.0502\times0+2.9223\)

Step2: Simplify the expression

Using the order of operations (multiplication first, then addition), \(-0.0502\times0 = 0\). So \(\hat{y}=0 + 2.9223=2.9223\). In the context of the problem, \(x = 0\) means the number of hours of video - game playing is \(0\) (a student who does not play video games), and \(y\) is the grade - point average.

Answer:

B. The grade - point average of a student who does not play video games is \(2.9223\)