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Question 3
4
Using the sample of 50 students from question #1, the variables of GRE and undergraduate GPA were used to predict graduate GPA. This regression analysis resulted in the following.
Multiple R = .750 R Square = .563 Standard Error = .066
The variable of gender was added to this set of predictors and the following resulted:
Multiple R = .770 R Square = .593 Standard Error = .064
3a. Did the addition of gender allow for a significant increase in the explained variance in graduate GPA?. Perform the appropriate test to address this question and briefly explain what it means.
3b. Test the overall regression equation (including the variables of GRE, undergraduate GPA, and gender) for statistical significance. Report the F value obtained from this test and briefly explain what it means.
Question 4
A researcher had a hypothesis that sense of humor was the key predictor of mental health. However, this researcher also believes that a belief in a team and having a support network also help your mental health. After consultation with the ERMA 7310 class, s/he ran an initial regression analysis and found the following.
1
Model Summary
5
Model
R R Square
Adjusted R Square
Std. Error of the Estimate
R Square Change
F Change
df1
df2
Sig. F Change
.975a .951
.945
1.08812
.951
185.690
3
29
.000
a. Predictors: (Constant), satsupport, AUfan, humor
Change Statistics
ANOVAb
Model
Sum of Squares df
Mean Square
F
Sig.
1
Regression
659.576 3
219.859
185.690
.000a
Residual
34.336 29
1.184
Total
693.912 32
a. Predictors: (Constant), satsupport, AUfan, humor
b. Dependent Variable: mental
4a. The null hypothesis is that these three predictors do not relate to the DV of Mental Health. Is there a relationship between these three predictors and the DV of mental health? Provide at least two sources of information to support your decision.
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Coefficientsa
Unstandardize d Coefficients
Standardized Coefficients
95.0% Confidence Interval for B
Correlations
Collinearity Statistics
Model
Std. B Error
Beta
t
Sig.
Lower Bound
Upper Bound
Zero-
order Partial Part
.969 .201 .046 .972 .422 .104 .959 .025 .006
Tolerance
1
(Constant)
-.368 .902
-.407 .687 1.107 .278 2.507 .018
.134 .894
-2.212
1.477
humor
.437 .395
.382
-.371
1.245
.014
69.726
AUfan
.378 .151
.562
.070
.686
.034
29.437
satsupport
.025 .190
.036
-.363
.414
.024
42.222
a. Dependent Variable: mental
VIF
4b. Examine the information summarized in the Coefficients table above. Which predictor variables (IVs) contribute to the prediction of the DV of Mental health? Provide at least three sources of evidence to support your decision.
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Regression #2
After having the initial analysis reviewed, the researcher was asked by the Board of Trustees and Booster Club to examine the effects of team support above and beyond social support and sense of humor. They wanted to tell alumni that team support was most important in hope that they would get larger alumni donations. The following results were obtained.
Model Summary
Change Statistics
Model 1
2
R .969a .975b
Adjusted R R Square Square
.940 .936
.951 .945
Std. Error of the Estimate
1.18009
1.08812
R Square
Change F Change df1
.940 234.141
.011 6.286
df2
Sig. F Change
.000 .018
2
1
30 29
a. Predictors: (Constant), humor, satsupport
b. Predictors: (Constant), humor, satsupport, AUfan
4c. Does team support (AUfan) contribute above and beyond humor and social support?
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Regression #3
Finally, the researcher was asked to just estimate whether being a fan had an effect and predict mental health. The following results emerged.
Model Summary
Model
R R Square
Adjusted R Square
Std. Error of the Estimate
R Square Change
F Change
df1
df2
Sig. F Change
.972a .945
.943
1.11373
.945
528.427
1
31
a. Predictors: (Constant), AUfan
Change Statistics
.000
1
Coefficientsa
Unstandardized Coefficients
Standardized Coefficients
95.0% Confidence Interval for B
Correlations
Zero-
order Partial Part
.972 .972 .972
Collinearity Statistics
Model
Std. B Error
Beta
t
Sig.
Lower Bound
Upper Bound
Tolerance
1
(Constant)
.955 .211
4.517 .000 22.988 .000
.524
1.387
AUfan
.654 .028
.972
.596
.712
1.000
a. Dependent Variable: mental
VIF 1.000
4e. What is the predicted mental health score of a person with a team support (AUfan) score of 8?
4d. What is the regression equation that would be used to predict mental health using team support as a predictor?

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3a. Did the addition of gender allow for a significant increase in the explained variance in graduate GPA?. Perform the appropriate test to address this question and briefly explain what it means.

We can use the R²∆ F-test compare the R² from this model and the full model derived earlier.

where: R²(L) = R² from the larger model = .593

k(L) = number of predictors in larger model = 3

R²(S) = R² from the smaller model = .563

k(S) = number of predictors in smaller model = 2

N = number of subjects 50

F = ((0.593-0.563))/(3-2)/((1-0.593)/(50-3-1)) = 0.030/0.008847826 =3.390663391

looking at an F-table F(1,50, α = .05) = 4.034309707 so, this R²-change is not significant at the .05 level because the F calculated value is smaller than the F tabulated value. It means that the addition of gender variable in the regression model is not needed as it does not help in predicting the dependent variable.

3b. Test the overall regression equation (including the variables of GRE, undergraduate GPA, and gender) for statistical significance. Report the F value obtained from this test and briefly explain what it means.

For overall model, we have

Multiple R = .770

R Square = .593

Standard Error = .064

R Square = .593

Regression/Total = 0.593

Regression = 0.593*Total...