Question

please use R for this problem, provide R codes.

Kiplinger's "Best Values in Public Colleges" provides a ranking of U.S. public colleges based on a combination of various measures of academics and affordability. The dataset "EX11-18BESTVAL.csv" includes a sample of 25 colleges from Kiplinger's 2015 report. Let's focus on the relationship between the average debt in dollars at graduation (AveDebt, the response variable) and the explanatory variables Admit (admission rate), GradRate (graduation rate), InCost Aid (in-state cost per year after need-based aid), and OutCostAid (out-state cost per year after need-based aid). a) (2 points) Write out the statistical model for this analysis, making b) (3 points) Run the multiple regression model in R and report the fitted regression equation. c) (5 points) State the null and alternative hypothesis for the overall F test, report the overall F statistic, its degrees of freedom, and the p-value. What do you conclude bnsed on this test result? d) (2 points) Obtain the residuals from part (b), construct a residual plot of residuals against the predicted outcome ý, and check assumptions. Is Baruch College an unusual case? Provide a brief summary. to specify all assumptions. sure multiple regression model but this time without Baruch College, and specify the e) (3 points) Run the same fitted regression equation. Again comment on the residuals (i.e., construct a residual plot of residuals against the predicted outcome ý and check assumptions). f) (5 points) For the model in part (e) (i.e., the multiple regression model without Baruch College), report the overall F statistic, its degrees of freedom, and the p-value. What do you conclude based on this test result? g) (5 points) For the model in part (e) that included all p 4 explanatory variables, only InCost Aid is found significant using the individual parametert tests. This raises the question whether these other three variables further contribute to the prediction of average debt given in-state cost is in the model. Conduct a partial F test to answer this question. Name State Admit GradRate TotalCostlr InCostAid TotalCostCOutCostAic AveDebt 20708 17468 University c FL University c CA University c WA North Caro NC The Univer TX The Ohio S OH 47 64 10359 39511 32637 17233 18 72 29640 6541 54270 31171 55 24433 58 10433 45552 34657 47042 31552 21263 47 41 51 19402 22036 9821 25076 38329 19530 25300 26472 23562 40 13323 56 59 21135 19366 12103 37635 28603 James Mac VA The Colleg NJ State Unive NY The Univer TX Baruch Col NY 61 43 53 65 26666 11806 34226 73 27901 20292 16265 38514 29942 42070 26878 32362 20790 67 24349 14699 59 46 39 43 22548 14465 33987 20504 27 23433 16306 33453 33906 26326 25354 30316 9949 The Univer AL 57 73 10854 19406 28508 St. Mary's (MD Michigan S M University c AR Florida Inte FL University c FL University cOK University c MA Oklahoma OK The Univer TN Colorado S CO 71 26754 15466 1438 41604 24624 25821 69 50 23414 19064 18591 17069 12300 45179 31154 36154 24361 25177 25139 12775 59 37 12271 12778 11421 24111 17893 17617 23964 28999 22541 23729 23726 25729 43 53 21 30990 30787 21 62 63 76 72 77 56 49 62 27 59 4735 16461 10669 15089 13251 14758 14466 17093 20340 41455 29767 42214 25900 32016 23254 33549 29431 26398 34 17182 23754 21525 21753 21071 37 38 37705 University NY State Unive NY The Richar NJ 56 33393 40 30721 31840 24116 26946 33944 42 25319 23614

Answer 1

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b) Multiple linear regression fit <- lm(Ave Debt admit GradRate + InCostAid OutCostAid, data=data) summary (fit) # shov results Call lm (formula = AveDebt admit + GradRate InCostAid OutCostAid, data data) Residuals: 30 1Q Median Маx Min 7520.2 -723.2 2004.0 -9510.6-2141.2 Coefficients: Estimate Std. Error t value Pr >t) 6477.7114 1.464 (Intercept) 9484.2748 0.1587 2.456 0.0233 admit 162.4033 66.1371 GradRate 64.8050 1.912 0.0703 123.8972 InCostAid 1.240 0.2295 0.3999 0.3226 0.2800 OutCostAid 0.1971 -1.110 -0.2199 0.05 '.'0.1 1 0.001 0.01 Signif. codes: 0 Residual standard error: 4395 on 20 degrees of freedom Adjusted R-squared: 0.2414 Multiple R-quared: 0.3678 p-value: 0.04771 E-3tatistic: 2.909 on 4 and 20 DF. Thus the regression model is 123.8972 x GradRate+0,3999 x InCost Aid-0.2189 x OutCostAid Ave Debt 9484.2748162.4033 x admit

c) F statistic BiB2=B3 = Ba=0 Hi:B 0 for at least 1j Ho B From the sunmary od model, we get F-statistic: 2.909 on 4 and 20 DF, p-value: 0.04771 As p-value is<0.05, we reject null hypothesis at 5% level of significance. Thus our model is better than model without any independent variables. d) Residual analysis fit$residuals 6 6062.66570 2003.990293 11 12 6228.63601 -6151.32351 -2606. 64230-9510.63021 564.25750 A 2 1 -1337.40434 347.09494 -1605.33269 -1105.65441 1:0 9 7 -2766.62811 Fennan 18, -775.53823 -214.204001968.238e6 23 24 9825036 A041.69335 174 16 15 14 13 1095.4679s 885.79469 -5061.58621 22 20 21 19 -723.23191 -1828.49761 2398.41380 -2675.011855 25 7520.19472 piot (fit)

Residuals vs Fitted 250 O 11 30000 26000 28000 24000 22000 20000 18000 Fitted values Im(AveDebt admit+ GradRate InCostAid + OutCostAid) 0009 0009: 0000 Residuals

From Residual vs Fitted plot we say that there are some exceptional cases like point 11, 25 and 8. We will check wthich college indices are these. According to data provided, 11th college is Brunch college, for which the residual vs fitted plot is having pure sign of some anomaly. e) Without Brunch college datal data [-c (11) , ] fit <lm (AveDebt admit + GradRate InCostAid Out CostAid, data-datal ) summary (fit) Call im (formula - AveDebt admit t GradRate +InCostAid OutCostAid, data datal) Residuals: 1Q Median Min 30 Max -724. -1663.2 -441.1 2219.5 5635.4 Coefficients: Estimate Std. Error t value Pr(>t |) (Intercept) 15656.1831 5515.2708 2.38.01051 0.50600 O17571 0.00939 -1.864 0.07791 admit 43.0909 63.5487 0.678 1.407 GradRate 75.9689 54.0107 InCcstAid 0.361 0.2391 2.892 OutCostAid C.1604 -0.2990 .05 Signif. codes: .01 .001 Residual standard errcr: 3539 on 19 degrees cf freedom Adjuated R-aquared: F-etatiatic: 3.661 on 4 and 1s F,p-value o.c2255 Multipie R-quared: 0.k359, plot it)

f) All is well This plot (TResidual vs fitted) is for checking the assumption of homoscedasticity. So if there is no conical pattern of scatter points, it is confirmed that the assurmption is valid. So now without Brunch college, the plot says the assumption is not voilated. F statistic p-value and DF are there in summary of fit again g) InCostAid is significant From summary, we see that tere are double star(asterisk) for InCostAid, which is only significant among all four independent variables.