Texas State University



Texas State UniversitySchool of Criminal JusticePh.D. Comprehensive Exam for StatisticsFebruary 24, 20159:00 a.m. – 11:00 a.m.DIRECTIONS: Choose Option One or Option Two.Save two electronic copies of your answer (one with just your ID number assigned to you, the other with your ID number and name). Email both copies to Cybele Hinson (ch56@txstate.edu). Print out and turn in a hard copy as well as with both your ID number and name on it.Option One:Background and MotivationAfter-school programs for juvenile students are thought to reduce a participant’s delinquency. These programs involve adult supervision, which is itself thought to reduce delinquency. Oftentimes, however, these programs provide activities that can reduce delinquency even later, when direct adult supervision is absent. Whether a student can be impacted by these programs may ultimately depend on the student’s age. Gottfredson et al. (2004), for example, found that after-school programs have a negative effect on delinquency for middle-school aged students, but not for elementary-aged students.This exam relies on the data from a sample of adolescent respondents. Estimate a multivariate ordinary least squares regression model to test the following hypothesis: Participation in after-school programs has a negative effect on delinquency, but the effect of participation on delinquency depends on age of the respondent. In other words, age moderates the effect that participating in an after-school program has on delinquency (such that this effect is stronger at higher ages).You may use a calculator. You will be assessed based on your responses to the following items:1.Using the data file described below, use SPSS to estimate a multivariate ordinary least squares regression equation. The dependent variable is a continuous measure of delinquency. The primary independent variables are: (1) a dummy-coded variable measuring whether the subject participates in an after-school program; (2) a continuous measure of the respondent’s age in years; and (3) the mathematical product of these two variables allowing for a statistical interaction between them. No variables here are centered on their mean.Hold constant the potentially confounding effects of the respondent’s: (1) time spent with peers; (2) religiosity; and (3) family attachment. No variables are centered on their mean.2.Interpret the model fit statistics for the model that you estimated.3.Based on the model you estimated, interpret and discuss (a) the y-intercept; (b) the slopes (i.e., the partial regression coefficients) for the primary independent variables; and (c) their tests of statistical significance. 4.After rounding to the nearest whole number, the measure of age in years has a mean of 15.0 and a standard deviation of 2.0. Based on these rounded quantities: (a) What is the effect of participation in an after-school program on delinquency when age is one standard deviation below the mean age? Report the actual numerical value, and whether this effect is statistically significant at the .05 level of statistical significance. Assume the standard error for the effect of participation in an after-school program on delinquency remains constant across ages.(b) What is the effect of participation in an after-school program on delinquency when age is one standard deviation above the mean age? Report the actual numerical value, and whether this effect is statistically significant at the .05 level of statistical significance. Assume the standard error for the effect of participation in an after-school program on delinquency remains constant across ages.(c) Explain whether and how these results support (or reject) the motivating hypothesis.5.Explain and discuss the error-term assumptions of the estimated model. Also, explain and discuss what the residuals from the estimated model indicate with regard to these assumptions. If evidence for problems exists, do not address problems with additional analysis.6.Explain and discuss: (1) collinearity; and (2) the estimated model’s assumptions regarding collinearity. Also, assess collinearity in the estimated model, and determine whether levels of multicollinearity are problematic for the model you estimated. If you determine that problems exist, do not attempt to address problems with additional analysis.The data file contains data from 400 individual respondents. The variables relevant to the exam are named and described below.Data File Contents for ExamVariable NameVariable Descriptiondelin =A continuous measure for delinquency. Higher values indicate more delinquency. Refer to units of this variable as points on the delinquency scale.program=A dummy-coded variable measuring whether the respondent participates in an after-school program.0 = No1 = Yesage=A continuous measure for the respondent’s age in years.product=The product-term for the statistical interaction between participating in an after-school program, and age (that is, product = program × age).timefrnd=A continuous measure of the respondent’s time spent with peers. Higher values indicate spending more time with peers. Refer to units of this variable as points on the time-spent-with peers scale.relig=A continuous measure of the respondent’s level of religiosity. Higher values indicate higher levels of religiosity. Refer to units of this variable as points on the religiosity scale.family=A continuous measure of the respondent’s level of attachment to their family. Higher values indicate higher attachment to family. Refer to units of this variable as points on the family-attachment scale.End of Option OneOption Two:Background and MotivationAfter-school programs for juvenile students are thought to reduce a participant’s delinquency. These programs involve adult supervision, which is itself thought to reduce delinquency. Whether a student can be impacted by these programs may ultimately depend on the student’s age. Gottfredson et al. (2004), for example, found that after-school programs have a negative effect on delinquency for middle-school aged students, but not for elementary-aged students.This exam relies on the data from a sample of adolescent respondents. The investigator tested the following hypothesis:Participation in after-school programs has a negative effect on delinquency, but the effect of participation on delinquency depends on age of the respondent. In other words, age moderates the effect that participating in an after-school program has on delinquency (such that this effect is stronger at higher ages).The primary independent variables are: (1) a dummy-coded variable measuring whether the subject participates in an after-school program; (2) a continuous measure of the respondent’s age in years; and (3) the mathematical product of these two variables allowing for a statistical interaction between them.The investigator used a multivariate ordinary least squares (OLS) regression model to test the hypothesis. The results of the estimation are presented in the table below. You may use a calculator. You will be assessed based on your responses to the following items:1. Interpret and discuss the model fit statistics.2.Interpret (a) the y-intercept; (b) the slopes (i.e., the coefficients) for the primary independent variables; and (c) their tests of statistical significance.3.The measure of age in years has a mean of 17.81 and a standard deviation of 1.95.(a) What is the effect of participation in an after-school program on delinquency when age is one standard deviation below the mean age? Report the actual numerical value, and whether this effect is statistically significant at the .05 level of statistical significance. Assume the standard error for the effect of participation in an after-school program on delinquency remains constant across ages.(b) What is the effect of participation in an after-school program on delinquency when age is one standard deviation above the mean age? Report the actual numerical value, and whether this effect is statistically significant at the .05 level of statistical significance. Assume the standard error for the effect of participation in an after-school program on delinquency remains constant across ages.(c) Explain whether and how these results support (or reject) the motivating hypothesis.4. The model’s total sum of squares is 3,116. The total sum of squares is notated with: (Yi-Y)2. The model’s explained sum of squares is notated with: (Yi-Y)2. What is the model’s explained sum of squares? Report the actual numeric value.5. Explain and discuss (1) collinearity; (2) the model’s assumptions regarding collinearity; and (3) whether levels of collinearity in this model are problematic.6. Explain and discuss the assumptions of the estimated model. Explain and discuss the desirable properties that this model (i.e., a multivariate OLS regression model) has when its assumptions are met.Table for statistics exam, option 2Ordinary least squares model explaining delinquencyN = 400VariablesCoefficientSEtVIFParticipant in after-school programa65.6391.230.7283.86Age in years6.414.501.424.42Participant in after-school programa × Age in years-3.315.12-0.6588.25Mathematical aptitude*-0.770.26-2.961.12Religiosity*-3.791.54-2.451.01Family attachment0.200.270.731.12Peer delinquency*12.852.854.501.06Malea*27.558.523.231.03Constant-76.5580.96-0.95----Model-fit statisticsR2=0.1247 Root MSE=83.528 Fdf1=8, df2=391,=6.96, p<.05* p < .05a A dummy-coded variable where zero indicates absence of characteristic.End of Option Two ................
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