Question

The data below represent commute times​ (in minutes) and scores on a​ well-being survey. Complete parts​ (a) through​ (d) below. Commute Time​ (minutes), x 5 15 30 40 60 84 105 ​Well-Being Index​ Score, y 69.1 68.0 66.8 66.1 64.9 64.1 62.0 ​(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.

Answer 1

`y=-0.065x+69.022`

Explanation:

The given data table is

Time​ (in minutes) x : 5 15 30 40 60 84 105

Well-Being Index​ Score y : 69.1 68.0 66.8 66.1 64.9 64.1 62.0

We need to find the least-squares regression line treating the commute​ time, x, as the explanatory variable and the index​ score, y, as the response variable.

The general form of least-squares regression line

`y=bx+a` .... (1)

where, a is y-intercept and b is slope.

`b=\frac{\sum_(i=1)^nx_iy_i-n\overline{x}\overline{y}}{\sum_(i=1)^nx_i^2-n\overline{x}^2}`

`a=\overline{y}-b\overline{x}`

Using graphing calculator we get

`b=0.0653469053052\approx 0.065`

`a=69.0218001284\approx 69.022`

Substitute the values of a and b in equation (1).

`y=-0.065x+69.022`

Therefore, the least-squares regression line is y=-0.065x+69.022.