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Linear and Multiple Regression Analysis

1. It is believed that GPA (grade point average, based on a four point scale) should have a positive linear relationship with ACT scores. Given below is the Excel output from regressing GPA on ACT scores using a data set of 8 randomly chosen UB students.

Regressing GPA on ACT

Regression Statistics
Multiple R 0.7598
R Square 0.5774
Adjusted R Square 0.5069
Standard Error 0.2691
Observations 8

df SS MS F Significance F
Regression 1 0.5940 0.5940 8.1986 0.0286
Residual 6 0.4347 0.0724
Total 7 1.0287

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 0.5681 0.9284 0.6119 0.5630 -1.7036 2.8398
ACT 0.1021 0.0356 2.8633 0.0286 0.0148 0.1895

(a) Write the regression equation
(b) Test the significance of independent variables
(c) Discuss the coefficient of determination
(d) Write the F-stat and discuss its meaning in this regression analysis

2. A weight-loss clinic wants to use regression analysis to build a model for weight-loss of a client (measured in pounds). Two variables thought to affect weight loss are client's length of time on the weight loss program and time of session. These variables are described below:

Y = Weight-loss (in pounds)
X1 = Length of time in weight-loss program (in months)
X2 = 1 if morning session, 0 if not
X3 = 1 if afternoon session, 0 if not (Base level = evening session)

Data for 12 clients on a weight-loss program at the clinic were collected and used to fit the interaction model:

Y = β0 + β1X1 + β2X2 + β3X3 + β4X1X2 + β4X1X3 + ε

Partial output from Microsoft Excel follows:

Regression Statistics
Multiple R 0.73514
R Square 0.540438
Adjusted R Square 0.157469
Standard Error 12.4147
Observations 12

F = 5.41118 Significance F = 0.040201

Coeff. StdError t Stat P-value
Intercept 0.089744 14.127 0.0060 0.9951
Length (X1) 6.22538 2.43473 2.54956 0.0479
Morn Ses (X2) 2.217272 22.1416 0.100141 0.9235
Aft Ses (X3) 11.8233 3.1545 3.558901 0.0165
Length*Morn Ses 0.77058 3.562 0.216334 0.8359
Length*Aft Ses -0.54147 3.35988 -0.161158 0.8773

(a) Write the regression equation
(b) Test the significance of independent variables
(c) Discuss the meaning of independent variables X1X2 and X1X3 (are they statistically significant? Explain.)

Solution Summary

This solution is comprised of a detailed explanation of various aspects of Regression Analysis as it pertains to the given problem. Supplemented with EXCEL output and more than 300 words of text, this step-by-step explanation of this complicated topic provides students with a clear perspective of Regression Analysis.