Question

Because elderly people may have difficulty standing to have their height measured, a study looked at the relationship between overall height and height to the knee. Here are data (in centimeters) for five elderly men: Knee Height ? 56.5 46.3 41.2 47.7 52.7 Height ? 192.4 151.8 149.7 162.1 174.5 What is the equation of the least-squares regression line for predicting height from knee height? ANSWER: ?̂ =

Answer 1

Solution:

The general regression equation is given as below:

y = a + b*x

Where a is y-intercept and b is the slope.

Formulas for a and b are given as below:

b = (∑XY – n*Xbar*Ybar)/(∑X^2 – n*Xbar^2)

a = Ybar – b*Xbar

The calculation table is given as below:

No.	Knee Height x	Height y	x^2	y^2	xy
1	56.5	192.4	3192.25	37017.76	10870.6
2	46.3	151.8	2143.69	23043.24	7028.34
3	41.2	149.7	1697.44	22410.09	6167.64
4	47.7	162.1	2275.29	26276.41	7732.17
5	52.7	174.5	2777.29	30450.25	9196.15
Total	244.4	830.5	12085.96	139197.8	40994.9
Mean	48.88	166.1

From above table, we have

n = 5

∑x = 244.4

∑y = 830.5

∑x^2 = 12085.96

∑y^2 = 139197.8

∑xy = 40994.9

Xbar = ∑x/n = 244.4/5 = 48.88

Ybar = ∑y/n = 830.5/5 = 166.1

b = (∑XY – n*Xbar*Ybar)/(∑X^2 – n*Xbar^2)

b = (40994.9 – 5*48.88*166.1)/( 12085.96 – 5*48.88^2)

b = 2.863954

a = Ybar – b*Xbar

a = 166.1 – 2.863954*48.88

a = 26.10993

y = a + b*x

y = 26.10993 + 2.863954*x

ANSWER: ?̂ = 26.10993 + 2.863954*x