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Class 28 Assignment AnswersThese questions refer to EMBS Case Problem 2. “Alumni Giving” which concerns data for 48 US national universities (America’s Best Colleges, Year 2000 Edition). Both the University of Notre Dame and the University of Virginia are included. The following five variables are in the data set.VariableSchool Graduation Rate% of Classes Under 20Student/Faculty RatioAlumni Giving RateDescriptionThe name of theUniversityPercentage of enrollees who graduatePercentage of Classes offered with <= 20 students.Number of students enrolled divided by total number of facultyPercentage of living alumni who gave to the University in 2000Mean?83.04255.72911.54229.271Median?83.559.510.529Mode?92651313Standard Deviation?8.60713.1944.85113.441Skewness?-0.282-0.5010.5820.370Minimum?662937Maximum?97772367Count?484848481. Test the hypothesis that graduation rate and alumni giving rate are (linearly) independent. We expect universities with higher graduation rates to have higher mean giving rates. [15 points]A regression of giving rate on graduation rate shows a positive linear relationship with reported p-value of 5.24E-10. For Ha: b>0, the p-value is half that, or 2.62E-10. We reject H0 in favor of Ha. The results are statistically significant.?CoefficientsStandard Errort StatP-valueIntercept-68.7612.58-5.461.82E-06 Graduation Rate1.180.157.835.24E-102. If the graduation rate of school A is 5 percentage points higher than that of school B, how much higher do we expect school A’s giving rate to be? [10 points]Using the above regression (graduation rate is all we know), the expected giving rate will be 1.18*5 = 5.9 percentage points higher for school A.3. If you learn that A and B above have identical student to faculty ratios, what is your revised answer to question 2? Be certain to explain why it went up (if it went up) or why it went down (if it went down) or why it stayed the same. Direct your response to a university administrator. [15 points] For this question, we know both graduation rate and student/faculty ratio. Since the latter is also predictive of giving rate, we will use a multiple regression to answer this question.?CoefficientsStandard Errort StatP-valueIntercept-19.1063115.55006-1.228700.22557 Graduation Rate0.755740.160234.716690.00002Student/Faculty Ratio-1.245950.28430-4.382500.00007(Note the p-value associated with student/faculty ratio is very low. Student/faculty ratio is an important variable which should not be ignored.) The 5 point higher graduation rate leads us to expect 0.756*5 = 3.8 percentage points higher giving rate for A. Our answer went down (5.9 to 3.8) because graduation rates and faculty/student ratios are negatively correlated in the sample. (Schools with higher graduation rates are expected to have lower faculty/student ratios….which in turn also lead to higher giving rates.) The answer to 2 reflected this reality. The higher grad rate for A would also imply a lower student faculty ratio…and the combination would lead to expecting 5.9 more percentage points in giving rate. When we learned that A did NOT have a lower student/faculty ratio than B, our expectations for its giving rate go down and we expect a smaller giving rate gap between the two schools.4. Provide a point forecast of alumni giving rate for a university with graduation rate of 80, 65 percent of its classes with 20 or fewer students, and a student/faculty ratio of 20. [25 points] (To answer this question, I expect you will build a linear regression model. Do not try anything fancy. Just pick which subset of the three numerically scaled variables you think comprise the best model.)From a modeling stand-point, the question is whether percent under 20 is needed. Does it add predictive poser to the model given we have both grad rate and student/faculty ratio? To see, we try the three-variable model.?CoefficientsStandard Errort StatP-valueIntercept-20.720117.5214-1.18260.2433 Graduation Rate0.74820.16604.50820.0000% of Classes Under 200.02900.13930.20840.8358Student/Faculty Ratio-1.19200.3867-3.08230.0035The p-value associated with %under20 is 0.83---not significant. We do not need and should not use all three variables. The model used to answer Q3 should be used to come up with the point forecast. Using a sumproduct to perform the calculation results in a point forecast of 16.4 for the alumni giving rate of the school in question. See below.?CoefficientsIntercept-19.10631 Graduation Rate0.75574Student/Faculty Ratio-1.24595??Intercept1 Graduation Rate80Student/Faculty Ratio20POINT FORECAST16.435. Of the 48 universities in the data set, which one has the most surprisingly low alumni giving rate? [10 points] (Hint: The answer is not U. of California-Davis. Its last-place giving rate is explained by its relatively low graduation rate and large classes.)I will use our 2-variable regression to calculate predictions (expectations) for each of the 48 schools and then identify the school with actual giving rate most below the prediction. This is the same thing as finding the school with the most negative residual.In the scatter plot of errors versus predicted, the circled point is the one with the most negative error. It is school 35 (U. of Michigan-Ann Arbor) for which the regression prediction was 24.9 but the actual giving rate was 13….a full 11.9 points below expectation. I will leave it to you Notre Dame readers to draw your own conclusions. (You can also identify the most negative residual by asking EXCEL to give you the residuals.....and either eyeball or sort.)6. Bo notices that some of the 48 have “university” in their names, some have “college” and the rest have “institute”. Bo wonders whether these names are predictive of student/faculty ratio? (Formulate and test a relevant hypothesis.) [25 points]Let us use H0: mean S/F ratio is equal for the three names. Ha will be not all equal. We can use either ANOVA single factor or regression with 2 dummies to test this hypothesis.SUMMARY OUTPUTRegression StatisticsMultiple R0.306267658R Square0.093799878Adjusted R Square0.053524317Standard Error4.719185001Observations48ANOVA?dfSSMSFSignificance FRegression2103.734851.86742.32900.1090Residual451002.181822.2707Total471105.9167????CoefficientsStandard Errort StatP-valueIntercept11.86360.711416.67540.0000Dcollege-0.36363.4120-0.10660.9156Dinstitute-7.36363.4120-2.15820.0363Although the mean giving S/F ratio for institutes is significantly lower than for Universities (the group not included in the model) because the p-value is 0.036, overall we CAN NOT reject H0 ( the p-value for our H0 is 0.1090). The differences in three sample means are not statistically significant. Part of the reason is that there are only 2 colleges and 2 institutes…which makes our estimates of their means highly uncertain---a fact accounted for in our p-value. ................
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