The data suggests that the distribution of percentage Alumni donation rate is normally distributed with estimated mean of 29 and standard deviation of 13.44. Scatter plot of Alumni giving rate and Graduation rate. The plot suggests a statistically significant [F(1,46) = 61.3, P < .001) positive correlation of .756 between graduation rate and alumni donation rate.
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The higher the graduation rate, the higher the alumni donation rate. The alumni donation rate ranged from 7% (University of California – Davis of CA) to 67% (Princeton University of NJ) with a mean of 29.3% (SD 13.4) whereas graduation rate ranged from 66% (University of Texas – Austin of TX) to 97% (Harvard University of MA) with a mean of 83% (SD 8.6). The regression equations giving the average relationship between the variables are: Alumni donation rate: -68.8 + 1.2 times graduation rate. I.e. Donation rate increases 1.2% on an average for every 1% increase in graduation rate. Variation in graduation rate could explain 57% variation in Alumni giving rate. Scatter plot of Alumni giving rate and percentage of classes under 20 The percentage of classes under 20 ranged from 29 (U. of Illinois-Urbana Champaign of IL) to 77 (Yale University of CT) with a mean of 55.7 (SD13.2). There is positive correlation (r = .646, F(1,46) = 32.9, P < .001) between the variables and the average relationship is expressed as: Alumni giving rate (%) = -7.4 + 0.66 times percentage of classes under 20. i.e. every 1% increase in percentage of classes under 20 increase the Alumni donation rate by 0.66%, on an average. Variation in percentages of classes under 20 explains 41.7% variation in Alumni donation rate. Relationship between Alumni donation rate and student/faculty ratio The Student/Faculty ratio ranged from 3 (California Institute of Technology of CA) to 23 (University of Florida of FL) with an average of 11.5 (SD 4.9). The observed correlation coefficient of -.742 does suggests a negative association between the variables at the population level [F(1,46) = 56.5, P < .001). Mathematical expression modeling the average relationship between the variables is: Alumni donation rate = 53 – 2.1 times Student/faculty ratio. Every unit increase in the student/faculty ratio brings down the alumni donation by 2.1%, on an average. Variation in student/faculty ratio could explain 55.1% variation is Alumni donation rate. Distribution of Alumni giving rate (%) in colleges from the East and the West The difference in the mean of percentage of Alumni donation rate in colleges from the East (n= 25, Mean = 35.0, SD = 11.7) and the West (n = 23, Mean = 23, SD = 12.5) is statistically significant (t(46) = 3.44; P = .001). Thus, the alumni from colleges in the East donate more often than their counterparts from colleges in the West. Question 2 The correlation between student/faculty ratio and Alumni giving rate (%) is strongly correlated (r = -.742). The estimated regression equation is: Alumni giving rate = 53.014 – 2.057 Â´ student/faculty ratio. Regression equation gives the mean of alumni giving rate for a given value of student /faculty ratio. Therefore, at a student/faculty ratio of 15, the average alumni giving rate = 53.014 – 2.057 Â´15 = 22.159. Standard error of this estimate is given by and has 46 (48 – 2) d.f. Here, sy.x = = 9.1 and x = 15. Hence SE = = 9.1 Â´ 0.178 = 1.62. [Since variance = 23.53, = 47Â´23.53)}. The question whether by keeping student/faculty ratio at 15, we could make the alumni giving rate at 50% is equivalent to testing whether the estimated mean alumni rate at student faculty ratio of 15 = 50 (H0: ÂÂ = ÂÂ0) against H1: ÂÂÂ¹50. Therefore the test statistic t = (22.159 – 50)/9.1 = -3.06 follows t distribution with 46 d.f. The two tailed probability (P-value) for |t| = 3.06 is .0037. Since the P-value is less than the conventional level of 5%, we reject the null hypothesis. i.e., by keeping student/faculty ratio at 15, we cannot make the alumni giving rate at 50%. Alternatively, the inference can be made from the scatter plot of alumni giving rate with student / faculty ratio with fitted regression line (average alumni rate for the student / faculty rate) and the 95% CI for the estimate. The value 50 for alumni giving rate is way outside the confidence interval at student/faculty ratio of 15. hence Prof Jimmy’s claim that alumni giving rte of 50% could be achieved by keeping student/faculty ratio at 15 could easily be rejected. In fact, even for the lowest student/faculty ratio reported in the study of 3, the estimated alumni giving rate is only 46.8%. The estimated alumni giving rates for student/faculty ratio of 5, 10, 15, 20 and 25 are 42.7, 32.4, 22.2, 11.9 and 1.6, respectively. Question 3 The location was coded as East = 1 and West = 0 for doing regression. The regression coefficient of location is the average difference in the alumni giving rate of East compared to West. The estimated regression equation is: Alumni giving rate = -16.47 + 0.689 Â´ graduation rate + 0.018 Â´ % of classes under 20 – 1.192 Â´ student/faculty ratio + 2.526 Â´ location. The contribution of graduation rate (+) and student/faculty ratio (-) towards alumni giving rate is statistically significant and all the variables together explains 70.7% variation in alumni giving rate.. Question 4
Alumni donations are an important source of revenue for privately funded Universities. It is important to study factors that could enhance the percentage of alumni who make a contribution and implement policies accordingly. Previous studies indicated that students with good contact with teachers have better chances of graduation. Prof Jimmy Michael, President of the University, collected data from 48 top Universities in US to study relationship between alumni giving rate (AGR) and factors like student/faculty ratio (SFR), % of classes under 20 (PC20), graduation rate (GR) and location of the University. The Universities in the East and West was almost equally represented in the sample (25 from East and 23 from West). The AGR ranged from 7% (University of California – Davis) to 67% (Princeton University) with a mean of 29.3% (SD 13.4). GR ranged from 66% (University of Texas – Austin) to 97% (Harvard University) with a mean of 83% (SD 8.6) and was positively correlated (r = .756) with AGR. Simple regression analysis suggested that AGR increases by 1.2% on an average for every 1% increase in GR and the contribution was statistically significant. PC20 ranged from 29% (U. of Illinois-Urbana Champaign) to 77% (Yale University) with a mean of 55.7 (SD13.2). The data showed a positive and statistically significant correlation (r = .646) with AGR. The contribution of PC20 to the AGR estimated from simple linear regression was 0.66% for every 1% increase in the PC20. SFR ranged from 3 (California Institute of Technology) to 23 (University of Florida) with an average of 11.5 (SD 4.9). The observed correlation coefficient of -.742 with AGR was statistically significant and simple regression analysis suggested every unit increase in the SFR brought down AGR by 2.1%, on an average. The data suggested that alumni from colleges in the East (mean AGR 35%, SD 11.7) donated more often than their counterparts from colleges in the West (mean AGR 23%, SD 12.5) and the difference was not a chance finding. The individual contribution of the four factors was assessed by multiple regressions. Statistically significant contributions came from GR and SFR. Higher GR and lower SFR increased AGR with similar relative importance. The estimated AGR (%) with SFR of 10 and GR (%) of 70, 75, 80, 85, 90 and 95 was 21.3, 25.1, 28.9, 32.7, 36.5 and 40.2, respectively. With an SFR of 15, the figures were 15.1, 18.9, 22.7, 26.4, 30.2 and 34.0, respectively. It may not be economically feasible to bring down the SFR beyond say, 10 and it is more important to increase the GR.
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