Question

according to a well - being survey, the well - being index composite score is comprised of six sub - indices life evaluation, emotional health, physical health, healthy behavior, work environment, and basic access. the data in the following table are based on the results of the survey, which represent commute time to work (in minutes) and the well - being index score. complete parts (a) through (d).
commute time (minutes), x: 5, 20, 30, 40, 50, 84, 105
well - being index score, y: 69.1, 67.8, 66.8, 66.3, 65.7, 64.3, 62.4
\\( \hat { y } = - 0.062 x + ( 69.003 ) \\)
(round to three decimal places as needed.)
(b) interpret the slope and y - intercept, if appropriate.
first interpret the slope. select the correct choice and, if necessary, fill in the answer box to complete your choice
a. for every unit increase in index score, the commute time falls by \\( \square \\), on average
(round to three decimal places as needed.)
b. for every unit increase in commute time, the index score falls by \\( \square \\), on average
(round to three decimal places as needed.)
c. for a commute time of zero minutes, the index score is predicted to be \\( \square \\)
(round to three decimal places as needed)
d. for an index score of zero, the commute time is predicted to be \\( \square \\) minutes
(round to three decimal places as needed )
e. it is not appropriate to interpret the slope

Explanation:

Step1: Recall the slope - intercept form of a linear regression equation

The linear regression equation is \(\hat{y}=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. In the given equation \(\hat{y}=- 0.062x + 69.003\), \(x\) represents commute time (in minutes) and \(\hat{y}\) represents the well - being index score.

Step2: Interpret the slope

The slope \(m=-0.062\). In the context of the regression equation \(\hat{y}=-0.062x + 69.003\), when \(x\) (commute time) increases by 1 unit (1 minute), \(\hat{y}\) (well - being index score) changes by the value of the slope. Mathematically, if \(x_1\) and \(x_2=x_1 + 1\), then \(\hat{y}_1=-0.062x_1+69.003\) and \(\hat{y}_2=-0.062(x_1 + 1)+69.003=-0.062x_1-0.062 + 69.003=\hat{y}_1-0.062\).

Answer:

B. For every unit increase in commute time, the index score falls by \(0.062\), on average.