Question

Both parts please, show work

a) Run a regression analysis on the following bivariate set of data with y as the response variable.

x	y
49.3	65.5
34.7	103.1
60.4	50.5
43.6	78.9
23.5	109.8
40.9	71.3
38.9	60.6
31.3	96.6
43	97.2
41.9	68.2
40.4	79.7

Verify that the correlation is significant at an ?=0.05?=0.05. If the correlation is indeed significant, predict what value (on average) for the explanatory variable will give you a value of 109.5 on the response variable.

What is the predicted explanatory value?
x = _____

b) Run a regression analysis on the following bivariate set of data with y as the response variable.

x	y
50.7	11.1
23.8	30.7
48.5	-3.6
19.5	34.7
12.1	48.3
31.8	21.7
47.6	16.3
25.7	39
20.6	43.4
46.8	8.5
51.1	16.5
33.8	17.9

Verify that the correlation is significant at an ?=0.05?=0.05. If the correlation is indeed significant, predict what value (on average) for the explanatory variable will give you a value of 23.4 on the response variable.

What is the predicted explanatory value?
x = _____

Answer 1

(a) the regression equation is given as y=146.02 - 1.62*x

for y=109.5,x=(109.5-146.02)/(-1.62)=22.54

the correlation between x and y =corr(x,y)=r=-0.8021 ,

we use t-test and

t =r/sqrt[(1—r²)/(n—2)]=(-0.8021)/SQRT((1-(-0.8021)*(-0.8021))/(11-2))=-4.03 with n-2=11-2=9 df

critical t(0.05,9)=2.26 is less than absolute value of calcuated t=4.03, so correlation coefficeitn is significant ( from zero)

following information has been generated using ms-excel

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.802120041
R Square	0.64339656
Adjusted R Square	0.603773955
Standard Error	12.05217637
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	2358.667221	2358.667	16.23812	0.002974517
Residual	9	1307.294597	145.255
Total	10	3665.961818
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	146.0170506	16.75014691	8.717359	1.11E-05	108.1255859	183.9085153
X Variable 1	-1.618190571	0.401570507	-4.02965	0.002975	-2.526606168	-0.709774974

(b) the regression equation y=58.73-x

for y=23.4, x=58.73-23.4=25.33

the correlation between x and y =corr(x,y)=r=-0.9109,

we use t-test and

t =r/sqrt[(1—r²)/(n—2)]=(-0.9109)/SQRT((1-(-0.9109)*(-0.9109))/(11-2))=-6.62 with n-2=11-2=9 df

critical t(0.05,9)=2.26 is less than absolute value of calcuated t=6.62, so correlation coefficeitn is significant ( from zero)

following information has been generated using ms-excel

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.910857071
R Square	0.829660603
Adjusted R Square	0.810734003
Standard Error	7.068498895
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	2190.192365	2190.192	43.83569	9.6924E-05
Residual	9	449.6730897	49.96368
Total	10	2639.865455
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	58.73062698	5.629001186	10.43358	2.51E-06	45.99694166	71.46431231
X Variable 1	-1.003270483	0.151531911	-6.62085	9.69E-05	-1.34605948	-0.660481487