c7 - Chapter 7 Regression Analysis Case Problem Alumni...

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Chapter 7Regression AnalysisCase Problem: Alumni Giving1.Descriptive statistics for graduation rate, % of classes under 20, student-faculty ratio, and alumni giving are shown below.GraduationRate% of ClassesUnder 20Student-FacultyRatioAlumniGiving Ratemean83.04255.72911.54229.271median83.559.510.529.0standard deviation8.60713.1944.85113.441minimum662937maximum97772367range31482060The correlations for each pair of variables are shown in the table below.GraduationRate% of ClassesUnder 20Student-FacultyRatioAlumniGiving RateGraduationRate1.00000.5828-0.60490.7559% of ClassesUnder 200.58281.0000-0.78560.6457Student-FacultyRatio-0.6049-0.78561.0000-0.7424AlumniGiving Rate0.75590.6457-0.74241.0000
2.The following Excel output provides the estimated simple linear regression model showing how thealumni giving rate (y) is related to the graduate rate (x).The estimated simple linear regression equation is ˆ68.76111.1805yx ., and the coefficient of determination r2is 0.5715, so this simple linear regression model explains approximately 57% of the variationin the sample values of alumni giving rate.Before using these results to test the hypothesis of no relationship between the alumni giving rate (y) and the graduation rate (x), we first check the conditions necessary for valid inference in regression. The Excel plot ofthe residuals and graduation rate follows.6065707580859095100-25025Graduation Rate Residual PlotGraduation RateResidualsThe residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badlyskewed at any value of displacement. Because there are no apparent severe violations of the conditions necessary for valid inference in regression, we will proceed with our inference. Since the level of significancefor use in hypothesis testing has not been given, we will use the standard 0.05 level throughout this problem.
The p-value associated with the estimated regression parameter b1is 5.23818E-10. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1= 0. We conclude that there is a relationship between the alumni giving rate and the graduation rate, and our best estimate is that a 1% increase in the graduation rate corresponds to an increase in the alumni giving rate of 1.1805%. The alumni giving rate is expected to increase as the graduation rate increases, so this result is consistent with what is expected.

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