Question

use the equation of the line of best fit, ( y = 3.87x + 15.52 ), to answer the questions below. give exact answers, not rounded approximations. (a) what is the predicted midterm score for a student who doesnt spend any time studying? (b) what is the predicted midterm score for a student who studies for 15 hours? (c) for an increase of one hour in the time spent studying, what is the predicted increase in the midterm score?

Explanation:

Step1: Solve part (a)

Substitute \(x = 0\) into \(y=3.87x + 15.52\).
\(y=3.87\times0+15.52\)
\(y = 15.52\)

Step2: Solve part (b)

Substitute \(x = 15\) into \(y=3.87x + 15.52\).
\(y=3.87\times15+15.52\)
First, calculate \(3.87\times15=(4 - 0.13)\times15=60-1.95 = 58.05\)
Then \(y=58.05+15.52=73.57\)

Step3: Solve part (c)

The equation of the line is in the form \(y=mx + b\) (slope - intercept form \(y=ax + c\) here \(a = 3.87\), \(b=15.52\)). The slope \(m\) (or \(a\) in our equation \(y = 3.87x+15.52\)) represents the change in \(y\) (mid - term score) for a unit change in \(x\) (time spent studying).
For a line \(y=mx + b\), if \(x_1\) and \(x_2=x_1 + 1\)
\(y_1=mx_1 + b\) and \(y_2=m(x_1 + 1)+b=mx_1+m + b\)
\(y_2-y_1=(mx_1+m + b)-(mx_1 + b)=m\)
Here \(m = 3.87\)

Answer:

(a) \(15.52\)
(b) \(73.57\)
(c) \(3.87\)