Question

1. The concept of fitting a line to bivariate data has been attributed to Francis Galton in an 1885 study of the heights of parents and their adult children. The table below presents the heights for a group of fathers and their adult sons. Create a scatter plot of the data. Find the least squares regression line of the son's height (y) on the father's height (x), and plot it on the scatter plot. Test the hypothesis that the slope of this line is equal to one. 68 Father's Height (in) Son's Height (in) 65 68 63 66 67 68 64 65 68 69 62 66 70 68 66 65 67 67 69 68 71 70 71

Answer 1

Excel > Data > Data analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.702651645
R Square	0.493719334
Adjusted R Square	0.443091268
Standard Error	1.403665056
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	1	19.21391076	19.21391076	9.751889966	0.010822249
Residual	10	19.70275591	1.970275591
Total	11	38.91666667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	35.82480315	10.17795313	3.519843598	0.005539879	13.14691036	58.50269594	13.14691036	58.50269594
X	0.476377953	0.15254826	3.122801621	0.010822249	0.136479248	0.816276657	0.136479248	0.816276657

a)

Scatter plot 72 71 70 69 Son's height (Y) 68 67 66 65 64 60 62 64 68 70 72 66 Father's height(x)

b)

Regression equation:

Y^ = 35.8248 + 0.4764*X

c)

Scatter plot 72 71 70 y = 0.4764x + 35.825 R = 0.4937 69 Son's height (Y) 68 67 66 65 64 50 62 64 66 68 70 72 Father's height(X)

d)

Hypothesis:

H0: β1 = 1

Ha: β1 not = 1

Test:

b1 = 0.4764, Sb1 = 0.1525

t stat = (b1-β1)/Sb1 = (0.4764-1)/0.1525 = -3.4334

P value = 0.0064

P value < 0.05, reject H0

There is enough evidence to conclude that slope of the regression line is not equal to 1