Question

A research studied the correlation between physical characteristics of sisters and brothers. Here are data on the heights (in inches) of 11 adult pairs.

Brother	71	69	66	67	70	71	70	73	72	65	66
Sister	69	63	65	63	65	62	65	64	66	59	62

a) Find the correlation and the equation of the least squares line for predicting sister’s height from brother’s height.

b) Carlos is 72 inches tall. Predict the height of his sister.

c) Based on the scatterplot and correlation r, do you expect your prediction to be very accurate? Why?

d) Is there evidence of a significant linear association between physical characteristics of sisters and brothers (a=0.05 level)?

e) What is the meaning of R2 in this example? Describe the meaning of R2 value in this example so that a non-statistician can understand.

Answer 1

Following table shows the caculations:

	Brother, X	Sister, Y	X^2	Y^2	XY
	71	69	5041	4761	4899
	69	63	4761	3969	4347
	66	65	4356	4225	4290
	67	63	4489	3969	4221
	70	65	4900	4225	4550
	71	62	5041	3844	4402
	70	65	4900	4225	4550
	73	64	5329	4096	4672
	72	66	5184	4356	4752
	65	59	4225	3481	3835
	66	62	4356	3844	4092
Total	760	703	52582	44995	48610

Sample size: n = 11

Now,

The coeffcient of correlation is :

Slope of the regression equation is

and intercept of the equation will be

So the regression equation will be

y'=26.8625+0.5362x

b)

The predicted value for x = 72 is

y'=26.8625+0.5362 *72 = 65.4689

Answer: 65.5 inches

c)

Following is the scatter plot:

Scatter plot and correlation coeffcient shows a linear positive and moderate relationship between the variables.

d)

Hypotheses are:

Degree of freedom: df=n-2=9

The critical value of r for correlation coefficient is: 0.602

Since r < 0.602 so relation between the variables is not significant.

That is there is no evidence of a significant linear association between physical characteristics of sisters and brothers.

e)

The R-square is:

That is 31.33% of variation in dependent variable Sister's height is explained by independent variable brother's height.