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

ANOVA
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

ANOVA
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.)