Question

Using software with the given​ data, find and interpret the multiple correlation and Upper R squared for the relationship between yequalscollege ​GPA, x 1equalshigh school​ GPA, and x 2equalsstudy time. Use both interpretations of Upper R squared as the reduction in prediction error and the percentage of the variability explained.

CGPA;HSGPA;Study Time
3.31;3.90;6
3.12;3.80;2
3.62;3.02;4
3.53;3.91;3
3.48;3.58;5
3.76;3.21;3
3.50;3.79;3
2.79;3.17;7
2.88;3.52;5
3.26;3.79;5
3.55;3.89;5
3.48;3.81;6
3.96;3.99;10
2.59;3.30;4
3.50;3.67;10
3.96;3.88;3
3.75;3.48;3
3.66;3.93;3
3.76;4.00;5
3.91;3.51;3
3.10;3.77;2
3.14;3.90;7
3.79;3.50;7
3.68;3.79;4
3.85;3.98;3
3.29;3.48;4
3.12;2.55;11
2.97;3.82;5
4.00;3.99;7
3.79;4.00;5
3.98;4.00;3
3.51;3.80;7
3.98;3.98;5
3.76;3.98;7
3.92;3.98;4
3.80;3.99;2
3.81;3.97;3
3.87;3.85;5
3.89;3.97;10
3.91;3.99;2
3.94;3.88;11
3.99;3.99;5
3.70;3.98;4
3.74;3.99;3
3.96;3.97;4
3.83;3.98;4
3.75;3.98;4
3.93;3.99;5
3.75;3.97;15
3.66;3.70;5
3.82;3.75;4
4.00;3.92;4
3.21;3.91;5
3.58;3.87;6
3.72;3.97;6
3.75;3.94;3
3.80;3.98;3
3.71;3.99;2
2.51;3.59;2

A) Identify the value of R^2​, as a​ decimal, from the output. Find the positive square root of R^2.

B) Find and interpret the multiple correlation and R^2 for the relationship between

y=college GPA, x1=high school​ GPA, and x2=study time. Use both interpretations of R^2

as the reduction in prediction error and the percentage of the variability explained.

Answer 1

Multiple Regressio output using EXCEL

let us assume that linaer equation will be

where y = College GPA, x1 = High School GPA, x2 = study time

A)

Regression Statistics
Multiple R	0.50
R Square	0.25
Adjusted R Square	0.23
Standard Error	0.32
Observations	59
ANOVA
	df	SS	MS	F	Significance F
Regression	2	1.94	0.97	9.57	0.000266
Residual	56	5.69	0.10
Total	58	7.63
	Coefficients	Standard Error	t Stat	P-value
Intercept	1.08	0.58	1.87	0.066972
HSGPA	0.65	0.15	4.37	5.42E-05
Study Time	0.01	0.02	0.76	0.452375

R² for the above given data for the equation from thge table is 0.25

and positive square root of R². = positive square root of (0.25) = 0.5

R² = 0.254 mean the model explains the 25% varaibiltiy of the data.

B) Multiple correlation (EXCEL Results)

	CGPA	HSGPA	Study Time
CGPA	1.000	0.497	0.024
HSGPA	0.497	1.000	-0.127
Study Time	0.024	-0.127	1.000

from the Multiple correlation results we can see that

correlation coefficient for CGPA vs HSGPA = 0.497

CGPA vs STUDY Time = 0.024 ( no linear relation ship)

HSGPA vs STUDY Time = -0.127 ( negative and and relation is not linear)

The correlation values between the variables (CGPA, HSGPA and Study time) are systematically low. This result indicates that the observed variables in each dataset do not share a large amount of variance.