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    Linear and Multiple Regression Analysis

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    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.)

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    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.

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