Question

the data below represent the number of days absent, x, and the final grade, y, for a sample of college students at a large university. complete parts (a) through (e) below.
no. of absences, x 0 1 2 3 4 5 6 7 8 9
final grade, y 87.9 85.1 82.2 79.8 76.9 72.5 63.0 67.6 64.8 61.9
(a) find the least-squares regression line treating the number of absences, x, as the explanatory variable and the final grade, y, as the response variable.
\\( \hat { y } = - 2.70 x + 82.32 \\)
(round to three decimal places as needed.)
(b) interpret the slope and y - intercept, if appropriate.
interpret the slope. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
(round to three decimal places as needed.)
a. for every unit change in the final grade, the number of days absent falls by days, on average.
b. for zero days absent, the final score is predicted to be 82.32.
c. for a final score of zero, the number of days absent is predicted to be days.
d. for every day absent, the final grade falls by, on average.
e. it is not appropriate to interpret the slope.
interpret the y - intercept. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
(round to three decimal places as needed.)
a. for zero days absent, the final score is predicted to be.
b. for a final score of zero, the number of days absent is predicted to be days.

Explanation:

Step1: Recall the slope - intercept form of a regression line

The regression line is of the form \(\hat{y}=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.

Step2: Interpret the slope

In the regression line \(\hat{y}=- 2.70x + 82.32\), the slope \(m=-2.70\). The slope represents the change in the response variable (\(y\), final grade) for a one - unit change in the explanatory variable (\(x\), number of absences). So, for every day absent (\(x\) increases by 1), the final grade (\(y\)) falls by \(2.70\) on average.

Step3: Interpret the \(y\) - intercept

The \(y\) - intercept \(b = 82.32\). In the context of the regression line \(\hat{y}=-2.70x + 82.32\), when \(x = 0\) (zero days absent), \(\hat{y}=82.32\). So, for zero days absent, the final score is predicted to be \(82.32\).

Answer:

Interpret the slope: D. For every day absent, the final grade falls by \(2.70\), on average.
Interpret the \(y\) - intercept: A. For zero days absent, the final score is predicted to be \(82.32\).