Possibly useful formulas for this exam: b1 = Corr(X,Y) SDY / SDX. confidence interval: Estimate ± (Critical Value) (Standard Error of Estimate)

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1 Statistics 5100 Exam 2 (Practice) Directions: Be sure to answer every question, and do not spend too much time on any part of any question. Be concise with all your responses. Partial SAS output and statistical tables are found in an accompanying handout. For any tests of significance, use α=0.05. The point value of each question is given, and the points sum to 100. Good luck! (Q1) (1 point) What is your name? Possibly useful formulas for this exam: b1 = Corr(X,Y) SDY / SDX b0 = Y b 1 X ei = Y i Y i confidence interval: Estimate ± (Critical Value) (Standard Error of Estimate) regression equation: Y i = β 0 + β 1 X i1 + β 2 X i2 + + β p 1 X i,p 1 + ε i (SSE reduced SSE full ) F p q = SSE full n p o p = # β s in full model (incl. intercept) o q = # β s in reduced model (incl. intercept) o n = sample size SSR(U V) = SSE(V) SSE(U, V) R 2 = SS model = 1 SSE SS total SS total 2 R adj = 1 n 1 SSE, where p = # predictors in model n p SS total First difference: Yt Yt-1 = Yt B Yt = (1 B) Yt (1 B) d Yt = (β0 + β1 Xt1 + + βk-1 Xt,k-1) + (1 ϕ1 B - ϕp Bp) -1 (1 θ1 B θq Bq) at o p : value at time t depends on previous p values o d : # of differences applied o q : value at time t depends on previous q random shocks Li = log π i = β 1 π 0 + β 1 X i1 + β 2 X i2 + + β p 1 X i,p 1 i OR j = e b j π i = 1/(1 + e L i)

2 Data: This exam will consider the following two data sets: I. The Out-of-state tuition data set, where variables in the following table are recorded on 109 public doctoral-granting (Type I) universities for the academic year. Variable Name Interpretation ostuit out-of-state tuition (in $'s) rbcost room and board costs (in $'s) sfratio student-to-faculty ratio palumdon percent of alumni who donate instexp instructional expenditure per student (in $'s) gradrate graduation rate fullcom average compensation for full professors (in $100's) fullsal average salary for full professors (in $100's) fullnum number of full professors high_ostuit indicator for high ostuit (1 if ostuit > 8,000; 0 otherwise) eastms indicator for being east of the Mississippi River (1 if yes; 0 otherwise) II. The Viscosity data set, where measurements have been taken on the viscosity of a chemical product known as XR-22 for 150 consecutive days. A chemical s viscosity is essentially its resistance to flow, or a measure of its consistency or texture. The variable daily is the only one in this data set. Of primary interest with this data set is better understanding the behavior of Daily over time, so that useful short-term predictions can be made.

3 Data Set I: Out-of-state tuition (Q2) A researcher suspects that, on average, universities with higher student-to-faculty ratios tend to have lower out-of-state tuition, so she regresses ostuit on sfratio (see partial SAS output for Model 1 ), assuming the following linear model: ostuiti = β0 + β1 sfratioi + εi (a) (6 points) In terms of the appropriate parameter(s) in this model, write out the null and alternative hypotheses of interest to this researcher. Also report the appropriate P-value. (b) (4 points) Based on this P-value, what can the researcher conclude regarding her suspicion, in the context of these data? (Be as specific as the null and alternative will allow.) (c) (6 points) Why can (or cannot) the researcher trust this conclusion? (Refer to the output.)

4 (Q3) (9 points) Referring to the P-value you reported in Q2 part a, what is the correct interpretation of this P-value, in the context of these data? (Not what conclusion would you reach based on its value [that s what you reported in Q2 part b], but what does the number itself actually mean?) (Q4) (4 points) In the Model 1 output, one plot refers to Cook s D. Explain clearly what this represents (and how it is used) in linear regression. (No credit will be given for formulas or specific numeric thresholds.) (Q5) (6 points) Model 2 involves ostuit being regressed on eight predictors. In the resulting output, which predictors, if any, appear to be collinear? Explain your response using specific numbers from the output. (Q6) (3 points; there is no SAS output for this question) Give one reason why multicollinearity is a problem in multiple linear regression. (For example, what will happen to the model if multicollinearity is not resolved?)

5 (Q7) (5 points) If you know there is collinearity involving two predictors, what does that tell you about their interaction in a linear regression model? (No output is provided for this question.) (Q8) (5 points) What proportion of the variation in out-of-state tuition is explained by its linear relationship with the eight predictors in Model 2? (Q9) (5 points) Model 3 involves ostuit being regressed on five predictors. Models 1-3 were fit using 80 randomly selected universities (of the 109 possible). The Model 3 fit was used to calculate predicted ostuit values for the remaining 29 universities, and their mean squared prediction error (MSPR) was calculated to be about 3.6 million (not shown in output). What can be concluded from this MSPR value? (Provide numerical justification.) (Q10) (6 points) One of the parameter estimates in the Model 3 output is reported as What is the correct interpretation of this number?

6 (Q11) (8 points) Model 4 is the logistic regression model used to predict the probability of high out-of-state tuition (high_ostuit=1) based on which side of the Mississippi River the university falls (eastms). (a) (6 points) What is the correct interpretation of the estimated eastms effect? (b) (2 points) What can be concluded from the plot following the Model 4 output? (c) (6 points) Based on the Model 4 fit, what is the predicted probability that Utah State University would have high out-of-state tuition? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Data Set II: Viscosity (Q12) (6 points) The Viscosity ARIMA Output reports the PROC ARIMA output for these data, with no predictor variables used. Referring to this output, what evidence is there of stationarity?

7 (Q13) (3 points) Referring again to the Viscosity ARIMA Output, what numerical evidence is there of significant dependence structure? (Q14) (8 points) Someone has proposed AR(2), MA(2), and ARIMA(2,0,1) models as possibly appropriate for these data. Based on the Viscosity ARIMA Output SAC and SPAC plots, which of these models appears most appropriate, and why? (Q15) (8 points) Each of the three proposed models are fit to the data (with partial output reported in the handout). For the model you selected in Q14, what can you say about its adequacy? (Refer to both numerical and graphical output.) (Q16) (1 point) What topic(s) did you study most that did not appear on this exam?

8 Output and Tables for STAT 5100 Exam 2 (Practice) Model 1 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model Error Corrected Total Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept <.0001 sfratio

9 Model 2 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept rbcost sfratio palumdon instexp gradrate fullcom fullsal fullnum Model 2 output continued on next page

10 Model 2 (continued) Collinearity Diagnostics Number Eigenvalue Condition Index Proportion of Variation Intercept rbcost sfratio palumdon instexp gradrate fullcom fullsal fullnum E E E E

11 Model 3 Number of Observations Read 80 Number of Observations Used 80 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept rbcost sfratio palumdon instexp gradrate <.0001

12 Model 4 Response Profile Ordered Value high_ostuit Total Frequency Probability modeled is high_ostuit=1. Convergence criterion (GCONV=1E-8) satisfied. Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi-Square Pr > ChiSq Intercept eastms Odds Ratio Estimates Effect Point Estimate 95% Wald Confidence Limits eastms

13 Viscosity ARIMA Output The ARIMA Procedure Name of Variable = daily Mean of Working Series Standard Deviation Number of Observations 150 Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq Autocorrelations < <

14 AR(2) model fit to Viscosity Data Unconditional Least Squares Estimation Parameter Estimate Standard Error t Value Approx Pr > t Lag MU < AR1, < AR1, Constant Estimate Variance Estimate Std Error Estimate AIC SBC Number of Residuals 150 Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq Autocorrelations AR(2) model fit output continued on next page

15 AR(2) model fit to Viscosity Data (continued)

16 MA(2) model fit to Viscosity Data Parameter Unconditional Least Squares Estimation Estimate Standard Error t Value Approx Pr > t Lag MU < MA1, < MA1, < Constant Estimate Variance Estimate Std Error Estimate AIC SBC Number of Residuals 150 Autocorrelation Check of Residuals To Lag Chi- Square DF Pr > ChiSq Autocorrelations MA(2) model fit output continued on next page

17 MA(2) model fit to Viscosity Data (continued)

18 ARIMA(2,0,1) model fit to Viscosity Data Parameter Unconditional Least Squares Estimation Estimate Standard Error t Value Approx Pr > t Lag MU < MA1, AR1, AR1, < Constant Estimate Variance Estimate Std Error Estimate AIC SBC Number of Residuals 150 Autocorrelation Check of Residuals To Lag Chi- Square DF Pr > ChiSq Autocorrelations ARIMA(2,0,1) model fit continued on next page

19 ARIMA(2,0,1) model fit to Viscosity Data (continued)

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