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. What proportion of variance in graduate grade point average can be accounted for by the variability in the quantitative GRE and verbal GRE scores in combination? (6 p)

2. Is this proportion of variance statistically significant at the .05 level of significance? Justify your answer. This includes providing the parameter being estimated, the statistical test (F, t, etc.) and its value, degrees of freedom associated with the statistical test. (14 p)

3. What proportion of variance in graduate grade point average cannot be accounted by the variability in the quantitative GRE and verbal GRE scores in combination? In other words, what is the unexplained variation in the regression model. (6 p)

4. Using an alpha level of .05, justify your decision as to whether the regression coefficients for GREQ and GREV (when evaluated separately or after controlling for the other) are statistically significant. Justifying your decision includes providing the parameter(s) being estimated, the statistical test (s) (t, F etc.) and its (their) value, degree(s) of freedom associated with the statistical test(s). (14 p)

Answer 1

I have used minitab to answer the following

First enter the data with corresponding header in 3 columns -> Stat -> Regression -> Regression -> Fit Regression model -> a dialogue box appears-> In 'Responses' enter GGPA -> In 'Continuous predictors' enter 'GREQ GREV' -> Ok -> the output appears in the session window as follows:

Regression Analysis: GGPA versus GREQ, GREV

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Regression 2 3.8344 1.91719 5.62 0.009
GREQ 1 1.6909 1.69088 4.96 0.035
GREV 1 0.4694 0.46944 1.38 0.251
Error 27 9.2131 0.34123
Lack-of-Fit 12 9.2131 0.76776 * *
Pure Error 15 0.0000 0.00000
Total    29 13.0475

Model Summary

S R-sq R-sq(adj) R-sq(pred)
0.584145 29.39% 24.16% 13.03%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant -1.32 1.28 -1.03 0.311
GREQ 0.00562 0.00252 2.23 0.035 1.28
GREV 0.00173 0.00148 1.17 0.251 1.28

Regression Equation

GGPA = -1.32 + 0.00562 GREQ + 0.00173 GREV

• R-sq for the regression is found to be 29.39% which implies 29.39% of variance in graduate grade point average can be accounted for by the variability in the quantitative GRE and verbal GRE scores in combination

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Regression 2 3.8344 1.91719 5.62 0.009
GREQ 1 1.6909 1.69088 4.96 0.035
GREV 1 0.4694 0.46944 1.38 0.251
Error 27 9.2131 0.34123
Lack-of-Fit 12 9.2131 0.76776 * *
Pure Error 15 0.0000 0.00000
Total    29 13.0475

• We can see that the p-value of the regression is 0.009<0.05 implying that the proportion of variance is statistically significant at 5% level.
• (100-29.39)% = 70.61% of the variance in graduate grade point average cannot be accounted by the variability in the quantitative GRE and verbal GRE scores in combination.

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant -1.32 1.28 -1.03 0.311
GREQ 0.00562 0.00252 2.23 0.035 1.28
GREV 0.00173 0.00148 1.17 0.251 1.28

From this table we can see that p-value of co-efficient of GREQ and GREV is 0.035<0.05 and 0.251>0.05

Hence the co-efficient of GREQ is statistically significant.