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
(a) find the least - squares regression line treating the commute time, x, as the explanatory variable and the index score, y, as the response variable.
\\( \hat { y } = \square x + \square \\) (round to three decimal places as needed.)

Explanation:

Step1: Calculate the necessary sums

Let \(n = 7\) (the number of data points).
Calculate \(\sum x\):
\(\sum x=5 + 20+30 + 40+50+84+105=334\)
Calculate \(\sum y\):
\(\sum y=69.1+67.8+66.8+66.3+65.7+64.3+62.4 = 462.4\)
Calculate \(\sum xy\):

$$
LATEXBLOCK0
$$

Calculate \(\sum x^{2}\):

$$
LATEXBLOCK1
$$

Step2: Calculate the slope \(b_{1}\)

The formula for the slope \(b_{1}\) of the least - squares regression line \(\hat{y}=b_{0}+b_{1}x\) is \(b_{1}=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\)
Substitute \(n = 7\), \(\sum x = 334\), \(\sum y = 462.4\), \(\sum xy=21695.7\), \(\sum x^{2}=23506\) into the formula:

$$
LATEXBLOCK2
$$

Step3: Calculate the intercept \(b_{0}\)

The formula for the intercept \(b_{0}\) is \(b_{0}=\bar{y}-b_{1}\bar{x}\), where \(\bar{x}=\frac{\sum x}{n}=\frac{334}{7}\approx47.714\) and \(\bar{y}=\frac{\sum y}{n}=\frac{462.4}{7}\approx66.057\)

$$
LATEXBLOCK3
$$

Answer:

\(\hat{y}=-0.049x + 68.395\)