Question

Refer to the data below. Suppose X is the independent variable and Y is the dependent variable. Calculate the variance of X, variance of Y, standard deviation of X, standard deviation of Y, the covariance between X and Y, the correlation coefficient between X and Y, the slope of the regression line, the Y intercept of the regression, ESS, RSS, TSS, and R-square of the regression line. Predict the value of Y when X=25. Show your work by constructing the 12-colume table.

X	Y
2	58
6	105
8	88
8	118
12	117
16	137
20	157
20	169
22	149
26	202

Answer 1

	X	Y	(x-x mean)	(y-y mean)	x-mean)^2	(y-mean)^2	(X-x mean)*(y- y mean)	Y-cap = 60+5x	ycap-y-mean	(y cap-y mean)^2	y-y cap	(y-y cap)^2	(y-y mean)^2
	2	58	-12	-72	144	5184	864	70	-60	3600	-12	144	5184
	6	105	-8	-25	64	625	200	90	-40	1600	15	225	625
	8	88	-6	-42	36	1764	252	100	-30	900	-12	144	1764
	8	118	-6	-12	36	144	72	100	-30	900	18	324	144
	12	117	-2	-13	4	169	26	120	-10	100	-3	9	169
	16	137	2	7	4	49	14	140	10	100	-3	9	49
	20	157	6	27	36	729	162	160	30	900	-3	9	729
	20	169	6	39	36	1521	234	160	30	900	9	81	1521
	22	149	8	19	64	361	152	170	40	1600	-21	441	361
	26	202	12	72	144	5184	864	190	60	3600	12	144	5184
mean=	14	130	0	0	56.8	1573	284	130	0	1420
sum=	140	1300	0	0	568	15730	2840	1300	0	14200	0	1530	15730

variance of X = (x-mean)^2 / (n-1) = 568/(10-1)=63.11

variance of Y=(y-mean)^2 / (n-1)=15730/9=1747.778

std dev of X = sqrt [ variance of X] = sqrt 63.11 =7.944

std dev of Y=sqrt [varince of Y] = sqrt 1747.778 = 41.806

Covariance(X,Y) = SUM(xi - xmean)*(yi - ymean)/(samplesize -1)=2840/9=315.5556

correlation coefficient,r= cov (X,Y) / (std dev of X*std dev of Y) = 315.5556/(7.944*41.806) = 0.950

slope = r*std dev of Y / std dev of X = 0.95*41.806/7.944=5

y-intercept = y mean - slope* x-mean = 130-5*14=60

ESS=sum (y cap-y mean)²=14200

RSS=sum(y - y cap)²=1530

TSS=ESS+RSS=14200+1530=15730

R-square = r^2 = 0.950^2 = 0.902734

or

R-square = ESS/TSS = 14200/15730=0.902734

when X=25,

predicted value of y =60+5*X = 60+5*25 = 185