Question

time spent studying (in hours)
use the equation of the line of best fit, $y = 3.82x + 15.72$, to answer the questions below.
give exact answers, not rounded approximations.
(a) for an increase of one hour in the time spent studying, what is the predicted increase in the midterm score?
(b) what is the predicted midterm score for a student who doesnt spend any time studying?
(c) what is the predicted midterm score for a student who studies for 12 hours?

Explanation:

Part (a)

Step1: Recall slope interpretation

The line of best fit is in the form \( y = mx + b \), where \( m \) is the slope (rate of change). Here, \( m = 3.82 \), so for a 1-unit (1 hour) increase in \( x \) (study time), \( y \) (midterm score) increases by \( m \).
<Expression> The slope \( m = 3.82 \) represents the predicted increase per 1-hour study time increase. </Expression>

Part (b)

Step1: Substitute \( x = 0 \) (no study time)

For \( x = 0 \), use \( y = 3.82x + 15.72 \). Substitute \( x = 0 \): \( y = 3.82(0) + 15.72 \).
<Expression> \( y = 0 + 15.72 = 15.72 \) </Expression>

Part (c)

Step1: Substitute \( x = 12 \) (12 hours study)

Use \( y = 3.82x + 15.72 \), substitute \( x = 12 \): \( y = 3.82(12) + 15.72 \).

Step2: Calculate \( 3.82 \times 12 \)

\( 3.82 \times 12 = 45.84 \).

Step3: Add 15.72

\( 45.84 + 15.72 = 61.56 \).
<Expression> \( y = 3.82(12) + 15.72 = 45.84 + 15.72 = 61.56 \) </Expression>

Answer:

s:
(a) \( 3.82 \)
(b) \( 15.72 \)
(c) \( 61.56 \)