Alumni giving case problem solution

Chapter 4 Regression Analysis

Case Problem: Alumni Giving

1. Descriptive statistics for graduation rate, % of classes under 20, student-faculty ratio, and alumni giving are shown below.

Graduation Rate

% of Classes Under 20

Student-Faculty Ratio

Alumni Giving Rate

mean

83.042

55.729

11.542

29.271

median

83.5

59.5

10.5

29.0

standard deviation

8.607

13.194

4.851

13.441

minimum

66

29

3

7

maximum

97

77

23

67

range

31

48

20

60

The correlations for each pair of variables are shown in the table below.

Graduation Rate

% of Classes Under 20

Student-Faculty Ratio

Alumni Giving Rate

Graduation Rate

1.0000

0.5828

-0.6049

0.7559

% of Classes Under 20

0.5828

1.0000

-0.7856

0.6457

Student-Faculty Ratio

-0.6049

-0.7856

1.0000

-0.7424

Alumni Giving Rate

0.7559

0.6457

-0.7424

1.0000

As would be expected, % of classes under 20 and student-faculty ratio have a relatively strong negative relationship (the correlation between these two variables is -0.7856). Using both of these variables as independent variables in a reg4ession model would introduce multicollinearity into the model.

The relationships the graduation rate has with % of classes under 20 and student-faculty ratio are both weaker.

2. The following Excel output provides the estimated simple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x).

The estimated simple linear regression equation is ., and the coefficient of determination r2 is 0.5715, so this simple linear regression model explains approximately 57% of the variation in 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 of the residuals and graduation rate follows.

The residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badly skewed 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 significance for 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 b1 is 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.

The estimated regression parameter b0 suggests that when the graduation rate is zero, the alumni giving rate is -68.7612%. This result is obviously not realistic, but this parameter estimate and the test of the hypothesis that 0 = 0 are meaningless because the y-intercept has been estimated through extrapolation (there is no university in the sample with a graduation rate near 0).

3. The following Excel output provides the estimated multiple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x1), % of Classes Under 20 (x2), and Student-Faculty Ratio (x3).

The multiple simple linear regression equation is ., and the coefficient of determination r2 is 0.6999, so this multiple linear regression model explains approximately 70% of the variation in the sample values of alumni giving rate.
Before using these results to test any hypotheses, we again check the conditions necessary for valid inference in regression. The Excel plots of the residuals and graduate rate, % of classes under 20, and student-faculty ratio follow.

The residuals appear to have a relatively constant variance and do not appear to be badly skewed at any value of displacement. However, the mean residual apears to deviate from zero at several values of graduation rate, suggesting a possible nonlinear relationship between the alumni giving rate and graduation rate. This does not appear to be severe, but we will keep this in mind as we continue.

The residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badly skewed at any value of displacement.

The residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badly skewed 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 significance for use in hypothesis testing has not been given, we will again use the standard 0.05 level throughout this problem.

The p-value associated with the F test for an overall regression relationship is 1.43233E-11. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 2 = 3 = 0 and conclude that there is an overall regression relationship at the 0.05 level of significance.

The p-value associated with the estimated regression parameter b1 is 4.799E-05. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 0 and conclude that there is a relationship between the alumni giving rate and the graduation rate. We estimate that holding % of classes under 20 and student-faculty ratio constant, a 1% increase in the graduation rate corresponds to an increase in the alumni giving rate of 0.7482%. The alumni giving rate is expected to increase as the graduation rate increases, so this result is consistent with what is expected.

The p-value associated with the estimated regression parameter b2 is 0.8358. Because this p-value is greater than the 0.05 level of significance, we do not reject the hypothesis that 2 = 0 and conclude that there is not a relationship between the alumni giving rate and the % of classes under 20 when controlling for the graduation rate and student-faculty ratio.

The p-value associated with the estimated regression parameter b3 is 4.799E-05. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 3 = 0 and conclude that there is a relationship between the alumni giving rate and the student-faculty ratio. We estimate that holding the graduation rate and % of classes under 20 constant, a 1 unit increase in the student-faculty ratio corresponds to decrease in the alumni giving rate of 1.1920%. The alumni giving rate is expected to decrease as the student-faculty ratio increases, so this result is consistent with what is expected.

The estimated regression parameter b0 suggests that when the graduation rate is zero, the alumni giving rate is -20.7201%. This result is obviously not realistic, but this parameter estimate and the test of the hypothesis that 0 = 0 are meaningless because the y-intercept has been estimated through extrapolation (there is no university in the sample with a graduation rate, a % of classes under 20, and a student-faculty ratio near 0).

4. Given our recognition in (1) of possible multicollinearity between the graduation rate and % of classes under 20, and the results from (2) and (3), it is reasonable to estimate a multiple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x1) and Student-Faculty Ratio (x2). The following Excel output provides this estimated multiple linear regression model.

The estimated multiple linear regression equation is ., and the coefficient of determination r2 is 0.6996, so removing the % of classes under 20 from the multiple linear regression estimated in (3) resulted in practically no loss in the ability of the regression model to explain variation in the sample values of alumni giving rate.
Before using these results to test any hypotheses, we again check the conditions necessary for valid inference in regression. The Excel plots of the residuals and the independent variables (graduate rate and student-faculty ratio) follow.

The residuals appear to have a relatively constant variance and do not appear to be badly skewed at any value of displacement. However, the mean residual apears to deviate from zero at several values of graduation rate, suggesting a possible nonlinear relationship between the alumni giving rate and graduation rate. This does not appear to be severe, but we will continue to keep this in mind as we continue.

The residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badly skewed 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 significance for use in hypothesis testing has not been given, we will again use the standard 0.05 level throughout this problem.

The p-value associated with the F test for an overall regression relationship is 1.76525E-12. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 2 = 0 and conclude that there is an overall regression relationship at the 0.05 level of significance.

The p-value associated with the estimated regression parameter b1 is 2.34782E-05. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 0 and conclude that there is a relationship between the alumni giving rate and the graduation rate. We estimate that holding student-faculty ratio constant, a 1% increase in the graduation rate corresponds to an increase in the alumni giving rate of 0.7557%. The alumni giving rate is expected to increase as the graduation rate increases, so this result is consistent with what is expected.

The p-value associated with the estimated regression parameter b2 is 6.95424E-05. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 2 = 0 and conclude that there is a relationship between the alumni giving rate and the student-faculty ratio. We estimate that holding the graduation rate constant, a 1 unit increase in the student-faculty ratio corresponds to decrease in the alumni giving rate of 1.2460%. The alumni giving rate is expected to decrease as the student-faculty ratio increases, so this result is consistent with what is expected.

The estimated regression parameter b0 suggests that when the graduation rate is zero, the alumni giving rate is -19.1063%. This result is obviously not realistic, but this parameter estimate and the test of the hypothesis that 0 = 0 are meaningless because the y-intercept has been estimated through extrapolation (there is no university in the sample with a graduation rate and a student-faculty ratio near 0).

Our results have suggested that the relationship between the alumni giving rate and the graduation rate may be nonlinear, so we will also estimate a model with a second order quadratic relationship between these two variables, i.e.,

where x1 is the graduate rate and x2 is the Student-Faculty Ratio. The following Excel output provides this estimated regression model.

The estimated multiple linear regression equation is ., and the coefficient of determination r2 is 0.7513, so adding the squared graduation rate as an independent variable resulted in an increase of 5% in the ability of the regression model to explain variation in the sample values of alumni giving rate.
Before using these results to test any hypotheses, we again check the conditions necessary for valid inference in regression. The Excel plots of the residuals and the independent variables (graduate rate and student-faculty ratio) follow.

The residuals appear to have a relatively constant variance and do not appear to be badly skewed at any value of displacement. Also note that the deviation of the mean residuals from zero at several values of graduation rate has been reduced.

The residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badly skewed 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 significance for use in hypothesis testing has not been given, we will again use the standard 0.05 level throughout this problem.

The p-value associated with the F test for an overall regression relationship is 2.38729E-13. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 2 = 3 = 0 and conclude that there is an overall regression relationship at the 0.05 level of significance.

The p-value associated with the estimated regression parameter b1 is 0.0093. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 0. Similarly, the p-value associated with the estimated regression parameter b2 is 0.0042. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 2 = 0. On the basis of the results of these two hypothesis tests, we conclude that there is a nonlinear relationship between the alumni giving rate and the graduation rate. We estimate that holding student-faculty ratio constant, a 1% increase in the graduation rate from a value of x1 to a value of x1 + 1corresponds to an increase in the alumni giving rate of

-6.9200 [(x1 + 1) – x1] + 0.0467 [(x1 + 1)2 –]
= -6.9200 (x1 – x1 +1) + 0.0467 (+ 2x1 + 1 –)
= -6.9200 + 0.0467 (2x1 + 1)

= -6.8733 + 0.0933x1

That is, estimated alumni giving rate initially decreases as graduation rate increases, and then eventually increases as graduation increases. Solving this result for x

-6.8733 + 0.0933x1= 0

-6.8733 = 0.0933x1

x1 = -6.8733 / - 0.0933 = 73.6532.

or 73.7%. The alumni giving rate decreases as the graduation rate increases until the graduation rate reaches 73.7%, at which point the alumni giving rate increases as the graduation rate increases. Perhaps alumni give more at low graduation rates alumni because they feel the university is in greater need of support, and they give more at high graduation rates because they are recognizing the university’s exceptional performance. If this explanation is reasonable, it also suggests that when the graduation rate is at an intermediate level, alumni are less motivated to give.

The p-value associated with the estimated regression parameter b3 is 6.68201E-06. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 3 = 0 and conclude that there is a relationship between the alumni giving rate and the student-faculty ratio. We estimate that holding the graduation rate constant, a 1 unit increase in the student-faculty ratio corresponds to decrease in the alumni giving rate of 1.3484%. The alumni giving rate is expected to decrease as the student-faculty ratio increases, so this result is consistent with what is expected.

The estimated regression parameter b0 suggests that when the graduation rate and the student-faculty ratio are zero, the alumni giving rate is 294.3311 %. This result is obviously not realistic, but this parameter estimate and the test of the hypothesis that 0 = 0 are meaningless because the y-intercept has been estimated through extrapolation (there is no university in the sample with a graduation rate and a student-faculty ratio near 0).

5. The residuals for the quadratic regression model, which were generated as part of the Excel output, are sorted and given in the table that follows.

University

Predicted Alumni Giving Rate

Residuals

U. of Washington

22.38665484

-10.3867

Stanford University

43.18507469

-9.1851

Columbia University

40.04079983

-9.0408

Johns Hopkins University

35.91193735

-8.9119

Northwestern University

38.69244742

-8.6924

Georgetown University

37.52122011

-8.5212

U. of Michigan–Ann Arbor

21.18859365

-8.1886

New York University

20.44978354

-7.4498

U. of Wisconsin–Madison

20.29550386

-7.2955

U. of Virginia

35.09496024

-7.0950

U. of California–San Diego

13.73838728

-5.7384

Harvard University

51.33050668

-5.3305

U. of California–Davis

12.14442958

-5.1444

College of William and Mary

31.86688013

-4.8669

Tulane University

21.79813595

-4.7981

Tufts University

33.32758156

-4.3276

Brown University

43.54883947

-3.5488

Washington University–St. Louis

34.87208824

-1.8721

Rice University

41.83672228

-1.8367

U. of California–Irvine

10.79607717

-1.7961

U. of Rochester

24.43764111

-1.4376

U. of California–Los Angeles

14.18214108

-1.1821

U. of Chicago

36.89270873

-0.8927

Boston College

25.72309551

-0.7231

Massachusetts Inst. of Technology

44.53342709

-0.5334

U. of California–Berkeley

18.49188883

-0.4919

U. of California–Santa Barbara

11.59983558

0.4002

U. of Texas–Austin

12.54835908

0.4516

U. of Pennsylvania

40.04079983

0.9592

U. of Southern California

21.03830244

0.9617

Cornell University

33.47616289

1.5238

Vanderbilt University

28.49978934

2.5002

Duke University

41.83672228

3.1633

Yale University

46.70262892

3.2974

Carnegie Mellon University

24.31196126

3.6880

Case Western Reserve Univ.

27.19154558

3.8085

Emory University

32.84765151

4.1523

Brandeis University

28.07130454

4.9287

U. of Illinois–Urbana Champaign

17.91487876

5.0851

California Institute of Technology

39.20661958

6.7934

U. of North Carolina–Chapel Hill

19.06132248

6.9387

Pennsylvania State Univ.

13.73838728

7.2616

U. of Florida

9.137455524

9.8625

Dartmouth College

42.65757169

10.3424

U. of Notre Dame

38.61251447

10.3875

Wake Forest University

25.80308452

12.1969

Lehigh University

25.11748561

14.8825

Princeton University

51.29809061

15.7019

These residuals suggest that the alumni giving rate several schools (U. of Florida, Dartmouth College, U. of Notre Dame, Wake Forest University, Lehigh University, and Princeton University) is at least 9% above what the model predicts. The Presidents of these universities should be pleased with the efforts of their Offices of Alumni Affairs.

These residuals also suggest that the alumni giving rate several schools (U. of Washington, Stanford University, Columbia University, Johns Hopkins University, Northwestern University, Georgetown University, U. of Michigan–Ann Arbor) is at least 8% below what the model predicts. Perhaps the Alumni Affairs offices at these universities should review the efforts of U. of Florida, Dartmouth College, U. of Notre Dame, Wake Forest University, Lehigh University, and Princeton University.

Other independent variables that could be included in the model include some measure of the success of the university’s football or basketball team and proportion of students on scholarship or student aid.

Graduation Rate Residual Plot

85 79 93 85 75 72 89 90 91 94 92 84 91 97 89 81 92 72 90 80 95 92 92 87 72 83 74 74 78 80 70 84 67 77 83 82 94 90 76 70 66 92 70 73 82 82 86 94 -6.5826771653543261 8.5004188306248949 -1.0268051599932875 14.417322834645674 8.2224828279443756 14.764030825934 -9.3047411626738068 -6.485257162003677 -3.6657731613335471 10.792678840676842 5.1537108393365827 6.597838833975544 -9.6657731613335471 0.25113084268721764 -9.3047411626738068 13.139386831965155 4.1537108393365827 -3.2359691740659997 -7.485257162003677 -4.6800971687049753 23.612162841346972 0.15371083933658269 -5.8462891606634173 -4.9437091640140665 0.76403082593400029 -11.221645166694586 -11.59700117272574 -9.59700117272574 -10.319065170045235 -17.680097168704975 -1.8749371754062594 5.597838833975544 8.6666108225833511 0.86145082928463523 -16.221645166694586 -2.0411291673647156 6.7926788406768424 3.514742837996323 2.0419668286145054 8.1250628245937406 3.8471268219132213 -11.846289160663417 -1.8749371754062594 -4.4164851733958699 2.9588708326352844 9.9588708326352844 0.23680683531580371 7.7926788406768424
Graduation Rate

Residuals

Graduation Rate Residual Plot

85 79 93 85 75 72 89 90 91 94 92 84 91 97 89 81 92 72 90 80 95 92 92 87 72 83 74 74 78 80 70 84 67 77 83 82 94 90 76 70 66 92 70 73 82 82 86 94 -3.5118498755356811 2.1750146512850321 -1.067219353946065 4.8129863288677939 2.5808078853063527 5.8769446183006835 -5.8708356564693958 -9.2760474840442271 1.0407119370650832 13.539578586852699 4.4486382596504015 1.3292119049748621 -8.012588477537733 1.5625210217678216 -9.9986400552124124 11.632209561994998 1.1517388159351469 -6.4824490411811837 -8.9969148451965779 3.5844128239127429 20.628057779017958 -0.37711785102956696 -7.7724130827572395 -6.5893964011594051 -3.471175197828785 -4.7992138162522053 -5.9264903799481985 -3.0248861679606165 -4.372598105541929 -9.8802375476073436 2.8807230848054033 -3.2468201775880559 16.107872640517932 3.148040389891726 -11.979950667087444 3.2796899568767586 13.347935855175663 0.84011510883579632 -3.0512123211701372 4.3043230397318908 8.2397088657391144 -5.8943327121312734 -6.4230372829356597 -6.4755749868175556 -0.87752305259686381 8.7678641697584681 -4.3994788794981972 6.4988961327697012
Graduation Rate

Residuals

% of Classes Under 20 Residual Plot

39 68 60 65 67 52 45 69 72 61 68 65 54 73 64 55 65 63 66 32 68 62 69 67 56 58 32 42 41 48 45 65 31 29 51 40 53 65 63 53 39 44 37 37 68 59 73 77 -3.5118498755356811 2.1750146512850321 -1.067219353946065 4.8129863288677939 2.5808078853063527 5.8769446183006835 -5.8708356564693958 -9.2760474840442271 1.0407119370650832 13.539578586852699 4.4486382596504015 1.3292119049748621 -8.012588477537733 1.5625210217678216 -9.9986400552124124 11.632209561994998 1.1517388159351469 -6.4824490411811837 -8.9969148451965779 3.5844128239127429 20.628057779017958 -0.37711785102956696 -7.7724130827572395 -6.5893964011594051 -3.471175197828785 -4.7992138162522053 -5.9264903799481985 -3.0248861679606165 -4.372598105541929 -9.8802375476073436 2.8807230848054033 -3.2468201775880559 16.107872640517932 3.148040389891726 -11.979950667087444 3.2796899568767586 13.347935855175663 0.84011510883579632 -3.0512123211701372 4.3043230397318908 8.2397088657391144 -5.8943327121312734 -6.4230372829356597 -6.4755749868175556 -0.87752305259686381 8.7678641697584681 -4.3994788794981972 6.4988961327697012
% of Classes Under 20

Residuals

Student-Faculty Ratio Residual Plot

13 8 8 3 10 8 12 7 13 10 8 7 10 8 9 11 6 13 8 19 5 8 7 9 12 17 19 20 18 19 20 4 23 15 15 16 13 7 10 13 21 13 12 13 9 11 7 7 -3.5118498755356811 2.1750146512850321 -1.067219353946065 4.8129863288677939 2.5808078853063527 5.8769446183006835 -5.8708356564693958 -9.2760474840442271 1.0407119370650832 13.539578586852699 4.4486382596504015 1.3292119049748621 -8.012588477537733 1.5625210217678216 -9.9986400552124124 11.632209561994998 1.1517388159351469 -6.4824490411811837 -8.9969148451965779 3.5844128239127429 20.628057779017958 -0.37711785102956696 -7.7724130827572395 -6.5893964011594051 -3.471175197828785 -4.7992138162522053 -5.9264903799481985 -3.0248861679606165 -4.372598105541929 -9.8802375476073436 2.8807230848054033 -3.2468201775880559 16.107872640517932 3.148040389891726 -11.979950667087444 3.2796899568767586 13.347935855175663 0.84011510883579632 -3.0512123211701372 4.3043230397318908 8.2397088657391144 -5.8943327121312734 -6.4230372829356597 -6.4755749868175556 -0.87752305259686381 8.7678641697584681 -4.3994788794981972 6.4988961327697012
Student-Faculty Ratio

Residuals

Graduation Rate Residual Plot

85 79 93 85 75 72 89 90 91 94 92 84 91 97 89 81 92 72 90 80 95 92 92 87 72 83 74 74 78 80 70 84 67 77 83 82 94 90 76 70 66 92 70 73 82 82 86 94 -3.9337995776850683 2.3708451220427094 -1.2094503837293686 4.6066651025372849 2.8856937590759699 5.6609928749287448 -6.2026946827405638 -9.1881977358988181 1.5317880626983396 13.526721286956729 4.5462850095400782 1.3462146237177777 -8.2060725332349591 1.7676080431929009 -9.9405552786738554 11.597234931437139 1.0543779455845481 -6.10923946518243 -8.9422442039210566 3.3205985805286851 20.541218233798475 -0.45371499045992181 -7.6996685224376833 -6.4290844921349901 -3.355192997160195 -4.4385146632351464 -6.1449890598547121 -2.8990355278769471 -4.4138841649102112 -9.6794014194713149 3.1239060452007834 -3.3916459722155139 16.128972820942383 2.6039906324259263 -11.930421727190677 3.0712671980565212 13.264581882890027 0.81180226410118195 -2.8700416341934627 4.4022313213564352 8.392801150256286 -6.2239473305710931 -6.8437222106213298 -6.8649748584518626 -0.65040752578782701 8.8414995381676995 -4.1652561628210876 6.7888606910234373
Graduation Rate

Residuals

Student-Faculty Ratio Residual Plot

13 8 8 3 10 8 12 7 13 10 8 7 10 8 9 11 6 13 8 19 5 8 7 9 12 17 19 20 18 19 20 4 23 15 15 16 13 7 10 13 21 13 12 13 9 11 7 7 -3.9337995776850683 2.3708451220427094 -1.2094503837293686 4.6066651025372849 2.8856937590759699 5.6609928749287448 -6.2026946827405638 -9.1881977358988181 1.5317880626983396 13.526721286956729 4.5462850095400782 1.3462146237177777 -8.2060725332349591 1.7676080431929009 -9.9405552786738554 11.597234931437139 1.0543779455845481 -6.10923946518243 -8.9422442039210566 3.3205985805286851 20.541218233798475 -0.45371499045992181 -7.6996685224376833 -6.4290844921349901 -3.355192997160195 -4.4385146632351464 -6.1449890598547121 -2.8990355278769471 -4.4138841649102112 -9.6794014194713149 3.1239060452007834 -3.3916459722155139 16.128972820942383 2.6039906324259263 -11.930421727190677 3.0712671980565212 13.264581882890027 0.81180226410118195 -2.8700416341934627 4.4022313213564352 8.392801150256286 -6.2239473305710931 -6.8437222106213298 -6.8649748584518626 -0.65040752578782701 8.8414995381676995 -4.1652561628210876 6.7888606910234373
Student-Faculty Ratio

Residuals

Graduation Rate Residual Plot

85 79 93 85 75 72 89 90 91 94 92 84 91 97 89 81 92 72 90 80 95 92 92 87 72 83 74 74 78 80 70 84 67 77 83 82 94 90 76 70 66 92 70 73 82 82 86 94 -0.72309550696339286 4.9286954625235566 -3.5488394710375601 6.7933804153491408 3.6880387428566053 3.8084544170177921 -4.8668801277106795 -9.0407998290606102 1.5238371113411944 10.342428307175538 3.1632777225822437 4.1523484904073129 -8.5212201119650501 -5.3305066825625218 -8.9119373510169169 14.882514391204584 -0.53342709295525026 -7.449783544138473 -8.6924474212918668 7.2616127225194269 15.701909387302877 -1.8367222774177563 -9.1850746851864997 -4.3275815571185205 -4.79813595190722 -0.49188883162657504 -5.1444295802872411 -1.7960771725184941 -1.1821410780332293 -5.7383872774805731 0.4001644181856534 -0.8927087328989245 9.862544475621899 5.0851212367116858 -8.188593647164069 6.9386775171787356 10.387485530481783 0.95920017093938981 -1.4376411077854243 0.96169756380443161 0.45164091738528001 -7.0949602385740178 -10.386654843964315 -7.2955038636667844 2.5002106627975138 12.196915478335004 -1.872088241263647 3.2973710838693009
Graduation Rate

Residuals

Student-Faculty Ratio Residual Plot

13 8 8 3 10 8 12 7 13 10 8 7 10 8 9 11 6 13 8 19 5 8 7 9 12 17 19 20 18 19 20 4 23 15 15 16 13 7 10 13 21 13 12 13 9 11 7 7 -0.72309550696339286 4.9286954625235566 -3.5488394710375601 6.7933804153491408 3.6880387428566053 3.8084544170177921 -4.8668801277106795 -9.0407998290606102 1.5238371113411944 10.342428307175538 3.1632777225822437 4.1523484904073129 -8.5212201119650501 -5.3305066825625218 -8.9119373510169169 14.882514391204584 -0.53342709295525026 -7.449783544138473 -8.6924474212918668 7.2616127225194269 15.701909387302877 -1.8367222774177563 -9.1850746851864997 -4.3275815571185205 -4.79813595190722 -0.49188883162657504 -5.1444295802872411 -1.7960771725184941 -1.1821410780332293 -5.7383872774805731 0.4001644181856534 -0.8927087328989245 9.862544475621899 5.0851212367116858 -8.188593647164069 6.9386775171787356 10.387485530481783 0.95920017093938981 -1.4376411077854243 0.96169756380443161 0.45164091738528001 -7.0949602385740178 -10.386654843964315 -7.2955038636667844 2.5002106627975138 12.196915478335004 -1.872088241263647 3.2973710838693009
Student-Faculty Ratio

Residuals

ˆ

68.76111.1805

yx

=-+

123

ˆ

20.72010.74820.02901.1920

yxxx

=-++-

12

ˆ

19.10630.75571.2460

yxx

=-+-

2

0112132

ˆ

ybbxbxbx

=++++

2

112

ˆ

294.33116.92000.04671.3484

yxxx

=-+-

2

1

x