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STAT 501 � Final Exam � Spring 2015

A final exam testing knowledge of statistical concepts and hypothesis testing.

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    1 STAT 501 – Final Exam – Spring 2015 1) ( 5x2 = 10 points) State which of the following statements are true and which are false. For the statements that are false, explain why they are false . a) In a logistic regression analysis where Y=1 represents survival and Y=0 represents death, t he logit of the survival probability is the negative of the logit of death probability. b) In regression analysis, t he method of ordinary least squares can be used in the presence of non - normal errors. c) In multiple linear regression analysis, t he width of a prediction interval for a future response of Y based on a single predictor X increases wi th the value of X . d) The error terms in an AR(1) model have zero mean. e) In model selection, the MSE (or S ) criterion minimizes confidence/prediction interval widths, while the PRESS criterion evaluates model unbiasedness. 2) ( 7x2 = 14 points ) Fill in the blanks with terms from the list: non - normality, multicollinearity, heteroscadasticity, confidence intervals, prediction intervals. [Note: there are 7 blanks but only 5 terms, so you’ll have to use some terms more than once and there may be some terms you don’t use at all .] a) A small p - value for the Ryan - Joiner tes t indicates non - normality. b) A residual vs . fits plot with a non - random pattern around a horizontal line at zero indicates heteroscedasticity. c) A large sample size ensures the validity of a confidence interval for a mean response eve n when e rrors exhibit non - normality d) Multicollinearity among predictors will lead to un reliable confidence intervals for regression coefficients . e) W eighted least squares estimation can be used in the presence of error heteroscadasticity and non - normality. 3) ( 4x 3 = 12 points) The following ANOVA table is abstracted from a regression fit to the model: Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + … + β 10 X 10 + ε . Source DF SS Regression 10 110.53 Residual Error 28 39.47 Total 38 150

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    2 Source DF Seq SS X1 1 0.10 X2 1 40 X3 1 1.0 X4 1 55 X5 1 2.5 X6 1 0.08 X7 1 6.5 X8 1 4.0 X9 1 0.4 X10 1 0.95 a) Calculate the three missing values in the upper table. b) For the 10 - predictor model , perform a hypothesis test at significance level 0.05 to determine whether predictors X 7 , X 8 , X 9 , and X 10 are significantly linearly related to Y upon controlling for the remaining predictors X 1 - X 6 using a general linear F test . Write the null and alternative hypotheses, the value of the test statistic, the decision rule, and the conclusion . [Note: an F - distribution table is provided on the last page of the exam.] c) Later it was decid ed to consider a regression of Y on the first 4 predictors ONLY . Use information from both tables above to calculate adjusted R 2 for the model with only the first 4 predictors . d) Given the information in both tables above , is it possible to test whether X 1 and X 3 can be dropped from the 4 - predictor model? Give a brief argument supporting your answer. [You do not have to do a test, even if one is possible.] 4) ( 4+9+4+4 = 21 points) Data from a local super market revealed that the deli usage of customers depe nds on the ir grocery bill and also on the time of shopping . To understand the link between these variables , a logistic regression model was fitted based on data from 890 sales records , which yielded the following. Source DF Adj Dev Adj Mean Chi - Square P - Value Regression 2 17.532 8.7660 17.53 0.000 bill 1 10.824 10.8241 10.82 0.001 lunch 1 5.549 5.5489 5.55 0.018 Error 887 534.290 0.6024 Total 889 551.822 Odds Ratio 95% CI Bill 1.0760 (1.0305, 1.1236) Odds Ratio for lunch=1 relative to lunch=0

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