A data set has $\bar{x}$ = 3, $\bar{y}$ = 10, S_x=1 S_y =3 r = 0.45 . Which of the following point is on the least-squares regression line? A) (10,3) B) (3,1) C) (1,3) D) (3,10)

Question

Quizplus · Accepted Answer

Using the formula for the equation of the least-squares regression line, we have:
$\hat{y} = b_1x +b_0$
where,
$b_1 = r \frac{S_{y}}{S_{x}}$
and
$b_0 = \bar{y} - b_{1}\bar{x}$
Plugging in the given values,
$b_1 = 0.45 \frac{S_{y}}{S_{x}}$
Also, for any point on the least-squares regression line, the coordinates should satisfy this equation. 
Thus, checking the options
A) (10,3): $\hat{y} = 0.45(\frac{S_{y}}{S_{x}})(10) + \bar{y} - 0.45(\frac{S_{y}}{S_{x}})(3) = 6.15\neq3$
B) (3,1): $\hat{y} = 0.45(\frac{S_{y}}{S_{x}})(3) + \bar{y} - 0.45(\frac{S_{y}}{S_{x}})(3) = 10\neq1$
C) (1,3): $\hat{y} = 0.45(\frac{S_{y}}{S_{x}})(1) + \bar{y} - 0.45(\frac{S_{y}}{S_{x}})(3) = 7.65\neq3$
D) (3,10): $\hat{y} = 0.45(\frac{S_{y}}{S_{x}})(3) + \bar{y} - 0.45(\frac{S_{y}}{S_{x}})(3) = 10$

Thus, only point D (3,10) is on the least-squares regression line.

A Data Set Has xˉ\bar{x}xˉ = 3 yˉ\bar{y}yˉ​ = 10, Sx=1 Sy =3 R = 0

A Data Set Has $\bar{x}$ = 3 $\bar{y}$ = 10, S_x=1 S_y =3 R = 0