Question

A researcher was interested in how students’ Graduate Record Examinations scores (GREQ- Quantitative and GREV-Verbal) predict college students’ graduate school Grade Point Average (GGPA). He collects data from 30 college students. The GRE Quantitative (X1) and GRE Verbal (X2) scores can range from 400-1600 (Note. This is the old GRE score scale). GGPA (Y) can range from 0.00 to 4.00.

GREQ	GREV	GGPA
625	540	2.16
575	680	3.60
520	480	2.00
545	520	2.48
520	490	2.88
655	535	3.44
630	720	3.68
500	500	2.40
605	575	3.76
555	690	2.72
505	545	2.96
540	515	2.08
520	520	2.48
585	710	2.16
600	610	4.00
625	540	2.16
575	680	3.60
520	480	2.00
545	520	2.48
520	490	2.88
655	535	3.44
630	720	3.68
500	500	2.40
605	575	3.76
555	690	2.72
505	545	2.96
540	515	2.08
520	520	2.48
585	710	2.16
600	610	4.00

1. If someone were to increase their GRE-Quantitative score by 50 points, how much would you expect his or her GPA to change, after controlling for the GRE-Verbal variability? Show how you calculated your answer. (8 p)

2. In your own words explain why you think that the model did not explain all of the variability in GGPA. In other words, what other factors might play a role in increasing the amount of variability explained in GGPA. (8 p)

3. Which of the two independent variables contributes more weight to the regression equation? In other words, which independent variable contributes more to the explanation of the dependent variable variability? Justify why you believe your answer is correct. (6 p)

4. Report and interpret the Pearson Correlation coefficients between GGPA & GREQ, GGPA & GREV, and GREQ & GREV. (12 p)

Answer 1

If someone were to increase their GRE-Quantitative score by 50 points, how much would you expect his or her GPA to change, after controlling for the GRE-Verbal variability? Show how you calculated your answer. (8 p)

The regression equation is:

Regression output					confidence interval
variables	coefficients	std. error	t (df=27)	p-value	95% lower	95% upper
Intercept	-1.3216
GREQ	0.0056	0.0025	2.226	.0346	0.0004	0.0108
GREV	0.0017	0.0015	1.173	.2511	-0.0013	0.0048

Predicted values for: GGPA
			95% Confidence Interval		95% Prediction Interval
GREQ	GREV	Predicted	lower	upper	lower	upper	Leverage
50	1	-1.03885	-3.45878	1.38108	-3.73934	1.66164	4.076

The GPA would be -1.03885 by putting GREQ = 50 and GREV = 1 in the regression equation.

In your own words explain why you think that the model did not explain all of the variability in GGPA. In other words, what other factors might play a role in increasing the amount of variability explained in GGPA. (8 p)

R²	0.294
Adjusted R²	0.242
R	0.542
Std. Error	0.584
n	30
k	2
Dep. Var.	GGPA

Only 29.4% of the variability is explained in CGPA. All the variability is not explained because the overall GRE score contribution is not included in the model.

Which of the two independent variables contributes more weight to the regression equation? In other words, which independent variable contributes more to the explanation of the dependent variable variability? Justify why you believe your answer is correct. (6 p)

The regression output with GREQ is:

	r²	0.258
	r	0.508
	Std. Error	0.588
	n	30
	k	1
	Dep. Var.	GGPA
ANOVA table
Source	SS	df	MS	F	p-value
Regression	3.3649	1	3.3649	9.73	.0042
Residual	9.6825	28	0.3458
Total	13.0475	29
Regression output					confidence interval
variables	coefficients	std. error	t (df=28)	p-value	95% lower	95% upper
Intercept	-1.1076
GREQ	0.0070	0.0022	3.119	.0042	0.0024	0.0116

Only 25.8% of the variability is explained in CGPA by GREQ.

The regression output with GREV is:

	r²	0.164
	r	0.405
	Std. Error	0.624
	n	30
	k	1
	Dep. Var.	GGPA
ANOVA table
Source	SS	df	MS	F	p-value
Regression	2.1435	1	2.1435	5.50	.0263
Residual	10.9040	28	0.3894
Total	13.0475	29
Regression output					confidence interval
variables	coefficients	std. error	t (df=28)	p-value	95% lower	95% upper
Intercept	0.9696
GREV	0.0033	0.0014	2.346	.0263	0.0004	0.0061

Only 16.4% of the variability is explained in CGPA by GREV.

Therefore, GREQ contributes more weight to the regression equation.

Report and interpret the Pearson Correlation coefficients between GGPA & GREQ, GGPA & GREV, and GREQ & GREV. (12 p)

Between GGPA & GREQ Pearson Correlation coefficient is 0.508. There is a moderate positive relationship between the variables.

Between GREQ & GREV Pearson Correlation coefficient is 0.468. There is a moderate positive relationship between the variables.

Between GGPA & GREV Pearson Correlation coefficient is 0.405. There is a moderate positive relationship between the variables.