Department of Mathematics The University of Toledo. Master of Science Degree Comprehensive Examination Applied Statistics.
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1 Department of Mathematics The University of Toledo Master of Science Degree Comprehensive Examination Applied Statistics April 8, 205 nstructions Do all problems. Show all of your computations. Prove all of your assertions or quote appropriate theorems. Books, notes, and calculators may be used. This is a three hour test.
2 . (30 points) A hospital administrator wished to study the relation between patient satisfaction (y) and patient's age (Xz, in years), severity of illness (x2, an index), and anxiety level (xa, an index). The administrator iandomly selected 46 patients and collected the data. (a). (5 points) Use Up criterion to choose the best subset of variables in the full model Y = /ÿlxl -ÿ- ÿ2x2 -t- /ÿ3z3 -ÿ- C and give this best model. (Use the best model to answer following questions) (b). (5 points) Find estimate of fl values in the chosen model from Up criterion. (c). (5 points) Obtain the studentized deleted residuals and identify any outlying y observations. Use the Bonferroni outlier test procedure with a =. 0. State the decision rule and conclusion. (d). (5 points) Obtain the diagonal elements of the hat matrix. dentify any outlying x observations. (e). (5 points) Give the estimate of mean patient satmfaction for patients who are Xl = 30 years old, whose index of illness severity is x2 = 58, and whose index of anxiety level is x Find the variance inflation factors. Do they indicate that a serious multieollinearity problem exists here? (f). (5 points) The two largest absolute studentized deleted residuals are for cases and 27. Obtain the DFFTS, DFBETAS, and Cook's distance values for this ease to assess its influence. What do you conclude? (Hint: F4,42 (0.5) = ) 2. (20 points) Refer to the attached table for "death penalty verdict by defendant's race and victims' race". The attached SAS output shows the results of fitting a logit model, treating death penalty as the response ( = yes) and defendant's race ( = white) and victims' race ( = white) as dummy predictors. (a). (5 points) nterpret parameter estimates. Which group is most likely to have the yes response? Find the estimated probability in that case. (b). (5 points) For a given defendant's race, find the 95% confidence intervals for conditional odds ratios of victim's race. (c). (5 points) Test the effect of defendant's race, controlling for victims'race, using (i) Wald test, and (i0 likelihood-ratio test. nterpret. (d). (5 points) Test the goodness of fit.
3 SAS OUTPUT Problem. data PatientSatisfaction; znput y xl x2 x3; datalines; , run; proc reg data=patientsatisfaction; model y = xl x2 x3/selection = cp b r vif influence; run; quzt; The RE6 Procedure Model: MODEL Dependent Variable: y C(p) Selection Method Number of Observatlons Read Number of Observatzons Used Number in Model C(p) R-Square ntercept Parameter Estimates xl x2 X , , , ,23 0, ,
4 The RE6 Procedure Model: MODEL Dependent Varlable: y Number of Observations Read Number of Observations Used Analyszs of Variance Source DF Sum o# Squares Mean Square F Value PP > F Model Error Corrected Total 2 9O38, , , <.OOOl Root MSE Dependent Mean Coeÿf Vat R-Square Ad] R-Sq Parameter Estimates Varzable DF Parameter Estimate Standard Error t Value Pr > tl Variance nflation ntercept xl x , , , ,75 <,000 <,O00 0,0086 o,
5 The RE6 Procedure Model: MOD L Dependent Varzable: y Output Statlstlcs Dependent Predicted Std Error Std Error Student Obs Varzable Value Mean Predict Reszdual Residual Reszdual Cook's D 48, , , ,8899,6657 4,90 9,896 8, ,5862 3,687 3, ,923 2,888 6,8077 9,63 0, , ,6889, , ,739, ,863 0, , , , , , , , , , , , , , , , , , ,0000 7, ,674 9, , , , ,435 9,390 0, , , , , , , ,280-4,0602 9,505-0, , ,6964-9, , ,9968 3,262 8, , , , , ,684 6, , , , , , , ,6530,7073 9,898, , , ,7462 2, , , , , , , , , , , , , ,63,020 * **N N* *** ** * ** N N **** * ** **l ** ** '* * ** ** ** **, ** 0,060 0, O.099 0, , , , ,002 0, ,05 0,027 0, ,029 0, ,03 0,069 0, ,
6 The REG Procedure Model: MODEL Dependent Varlable. Output Statlstzcs Hat Dzag Cov Obs RStudent H Ratio DFFTS... DFBETAS- ntercept xl x , , , , ,7040 0, ,20 0,904-0, ,2689 0, ,08-0, ,2968 0,0343 0, ,0827-0,365 0, ,2267 0, ,266 i , ii , , , , , , , , , , , ,2079-0, ,33-0,29 2 0,4459 0, , , , , ,900-0, , , ,i , ,0377-0,80 0, , , , ,8390 0, ,2839-0,27-0,208 0, , ,4262 0, , , , , , , , , ,359 0, , , , , , , , ,0373 4,032 0, , , ,80-0, , , , , , , , , ,86
7 Problem 2. Defendant's Death Penalty Percent Race Yes No Yes White White 53 4[4 3 Black Black White Black Total White Black Crzterza For Assesszng 6oodness Of Fzt Crzterzon DF Value Value/DF Devzance Scaled Devzance Pearson Chl-Square Scaled Pearson X2 0,978 Log Llkelzhood Full Log Likelihood AC (smaller is better) 9,2998 ACC (smaller is better) 8C (smaller zs better) Analyszs Of Maxzmum Likelzhood Parameter Estzmates Parameter DF Standard Lkellhood Ratzo 95% Nald Estlmate Error Confzdence Limits Chz-Square Pr > ChiSq ntercept VlCtlm defendant -3, ,7754-2, , ,03-8, , <,OOOl <.6OOl 0.08 LR Statistics For Type 3 Analysts Chi- Source DF Square Pr > ChlSq victim <,888 defendant
8 3. A recent study of undergraduates looked at gender differences in dieting trends. There were 8 women and 05 men who participated in the survey. The following table summarizes whether a student tried a low-fat dlet or not by gender: Gender Tried a low-fat die Women Men Yes 35 8 No (a) Fill in the missing cells of the table, (b) Summarize the data numerically and graphically, (c) 'rest that there is no association between gender and the likelihood of trying a low-fat diet. Summarize the results. Use a = 0,05, 4. Suppose the results of an experiment are as follows: Treatment group Control group (a) Calculate the difference in means between the two groups. (b) Write out all possible permutations of these observations to the two groups and calculate the difference in means. (c) What proportion of the differences are as large or larger than the observed difference in mean times? What is the exact P-value? (d) Summarize the results. Use cÿ = 0.05.
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