Question

Consider the following set of ordered pairs. 5 4 1 n 5 5 3 2 4 3 у 3 a) Calculate the slope and y-intercept for these data. b) Calculate the total sum of squares (SST). c) Partition the sum of squares into the SSR and SSE. a) Calculate the slope and y-intercept for these data. X у (Round to four decimal places as needed.) b) Calculate the total sum of squares (SST). SST = (Round to one decimal place as needed.) c) Partition the sum of squares into the SSR and SSE. SSE = (Round to three decimal places as needed.) (Round to three decimal places as needed.) SSR=

Answer 1

Solution-:

By using R-Software:

> x=c(5,5,3,4,1);x
[1] 5 5 3 4 1
> y=c(4,5,2,3,-3);y
[1] 4 5 2 3 -3
> # (a) For the equation of the line of regression
> fit=lm(y~x);fit

Call:
lm(formula = y ~ x)

Coefficients:
(Intercept) x
-4.357 1.821

> cr=coef(fit);cr
(Intercept) x
-4.357143 1.821429
> mcr=matrix(cr);mcr
[,1]
[1,] -4.357143
[2,] 1.821429
> a=mcr[1,1];a
[1] -4.357143
> round(a,4)
[1] -4.3571
> b=mcr[2,1];b
[1] 1.821429
> round(b,4)
[1] 1.8214
> #The equation of the line of regression of y on x is: y= m*x+b or y=a+b*x
> #The equation of the line of regression of y on x is:y=-4.3571+1.8214*x
> #(c) For SSE
> esty=-4.3571+1.8214*x;esty # Regression equation
[1] 4.7499 4.7499 1.1071 2.9285 -2.5357
> d=y-esty
> d1=d^2
> SSE=sum(d1);SSE #Residual Sum of Square
[1] 1.642857
> round(SSE,3)
[1] 1.643
> #For SSR
> my=mean(y);my #Mean of y
[1] 2.2
> c=esty-my
> c1=c^2
> SSR=sum(c1);SSR #Regression sum of square
[1] 37.15598
> round(SSR,3)
[1] 37.156
> #(b)For SST
> SST=SSR+SSE;SST #Total sum of square
[1] 38.79883
> round(SST,1)
[1] 38.8

R-Code:

x=c(5,5,3,4,1);x
y=c(4,5,2,3,-3);y
# (a) For the equation of the line of regression
fit=lm(y~x);fit
cr=coef(fit);cr
mcr=matrix(cr);mcr
a=mcr[1,1];a
round(a,4)
b=mcr[2,1];b
round(b,4)
#The equation of the line of regression of y on x is: y= m*x+b or y=a+b*x
#The equation of the line of regression of y on x is:y=-4.3571+1.8214*x
#(c) For SSE
esty=-4.3571+1.8214*x;esty # Regression equation
d=y-esty
d1=d^2
SSE=sum(d1);SSE #Residual Sum of Square
round(SSE,3)
#For SSR
my=mean(y);my #Mean of y
c=esty-my
c1=c^2
SSR=sum(c1);SSR #Regression sum of square
round(SSR,3)
#(b)For SST
SST=SSR+SSE;SST #Total sum of square
round(SST,1)

Or

By using MS-Excel :

Given Data:

x	y
5	4
5	5
3	2
4	3
1	-3

Output:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9786
R Square	0.957658
Adjusted R Square	0.943544
Standard Error	0.740013
Observations	5
ANOVA
	df	SS	MS	F	Significance F
Regression	1	37.15714	37.15714	67.85217	0.003746
Residual	3	1.642857	0.547619
Total	4	38.8
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-4.35714	0.862089	-5.05417	0.014944	-7.10069	-1.61359	-7.10069	-1.61359
x	1.821429	0.221121	8.237243	0.003746	1.117722	2.525135	1.117722	2.525135

From this output we get,

(a) The equation of the line of regression of y on x is:y=-4.3571+1.8214*x

(c) SSE=1.643 and SSR=37.156

(b) SST=SSR+SSE=37.156+1.643=38.8