Problem 1:

Background: A sociologist investigating the recent increase in the incidence of homicide throughout the United States studied the extent to which the homicide rate per 100,000 population \((Y)\) is associated with the city's population size (in thousands) \(\left(X_{1}\right)\), the percentage of families with yearly income less than \(\\) 5,000\left(X_{2}\right)$, and the rate of unemployment \(\left(X_{3}\right)\). Data are for a hypothetical sample of \(n=20\) cities.

1. Regress \(Y\) on \(X_{1}, X_{2}\), and \(X_{3}\). Provide and examine a plot of the residuals by the predicted values. Is there anything noteworthy about this plot?
2. Regress \(Y\) on \(X_{1}, X_{2}\), and \(X_{3}\). Provide and examine the leverage values and make a determination regarding the resultant values.
3. Regress \(Y\) on \(X_{1}, X_{2}\), and \(X_{3}\). Do you find any significant evidence that the normality assumption is violated? Provide evidence.
4. Regress \(Y\) on \(X_{1}, X_{2}\), and \(X_{3}\). Provide and examine the collinearity statistics and comment on whether or not there are any concerns. If any concerns arise, discuss how you might address them (but don't actually do it).

Problem 2 :

An investigator used the Theory of Reasoned Action (TRA) as a framework to assess factors that might help to predict students' intentions to read about statistics outside of class. The TRA suggests that behavioral intentions are a function of one's attitudes toward the behavior and one's subjective norms, which refer to normative pressure that one perceives to (or not to) perform a behavior.

Our investigator conducts a study with a random sample of $n=400$ students in introductory statistics classes (at the graduate level) at a variety of universities (ignore the possible violations of independence). Each construct (i.e., intentions, attitudes, subjective norms) was measured using a multi-item scale (a direct measurement approach was used, for those of you familiar with the TRA). Individual items were measured with a seven-point Likert-type scale (scored 0 to 6). Items for each construct were averaged to create a single variable representing each construct (with higher scores indicating stronger intention, more favorable attitudes, and greater subjective norms, respectively).

The investigator wants to know whether attitudes towards the behavior and subjective norms have equal or differential effects in the prediction of intentions in this context (this question is occasionally asked by researchers using this theoretical framework). Help the investigator assess whether these two variables have the same effect. Use \(\alpha=0.05\) for your test.

You can use SPSS or SAS (or any software for that matter) to conduct your analysis. Please address the following (you do not need to worry about diagnostics):

1. Write out the null and alternative hypotheses;
2. Calculate the test statistic;
3. Give the value of the critical value and state your rejection region;
4. State your decision;
5. Write your conclusion as you would if you were writing the results of this analysis for publication in a journal.

Problem 3 :

This problem uses an SPSS data set called

The data come from \(n=506\) communities surrounding a major metropolitan area. The dependent variable, lprice, is the natural logarithm of the median price of a single-family home in each community. The predictor variables of interest include rooms (the average number of rooms per house), lnox (the natural logarithm of nitrous oxide in parts per \(100 \mathrm{~m}\); this is a measure of air pollution), ldist (the natural log of the weighted distance to five employment centers), stratio (the average student-teacher ratio in local schools), and crime (crimes committed per capita). Regress the dependent variable on the set of all five predictor variables and then do the following:

1. Test for heteroscedasticity (or heteroskedasticity) in SPSS using the Breusch-Pagan test.
2. Read about the White test for heteroskedasticity in the Wooldridge excerpt. DO NOT actually perform this test, but instead tell me how many additional regressors (predictors) will be included in the White test relative to the Breusch-Pagan test in the present example. Show your work.
3. Compute heteroskedasticity-robust standard errors using SPSS (sometimes called White, Huber, or Huber-White standard errors). As discussed in class, the process of finding these standard errors is not automated in the SPSS Linear Regression procedure, but you can nevertheless use SPSS to calculate them using other procedures. I am attaching a technote from IBM SPSS that you can use to answer this question. It provides multiple options for finding the heteroskedasticity-robust standard errors - you can use any of them for this problem, but just use one of them and tell me which one you are using. Provide the heteroskedasticity-robust standard errors and write up a brief paragraph summarizing your observations concerning the findings. You must do this problem in SPSS, although you can confirm your results in SAS or STATA if you want to (but you do not have to).
4. Notice that the dependent variable in this model is log-transformed. Please read the attached handout entitled, "Handout 4: Coefficient Interpretation and Statistical Significance" (this is from an economics class at Iowa State University). Based on this reading, interpret the parameter estimates for 1) rooms and 2) ldist. One sentence for each variable should be plenty.

Problem 4 :

1. The method now known as the "jackknife" was given that name by John Tukey. Who is generally credited with being the first to propose this method?

Use the following tables to answer questions 2-6 (Hint: You may need to use the formulas on pages 172-173 of Kleinbaum et al. to answer these questions. The slides from Lecture 10 might also be of some help). Please show your work in the appropriate space.

The data that generated the above information come from 121 countries.

Assume you are trying to predict LIFEEXP using #DOCs and #RADIOs as the predictor variables using standard multiple regression.

3. What is the variation in LIFEEXP uniquely explained by #RADIOs? (Provide an actual value.)

4. What is the variation in LIFEEXP explained jointly by #DOCs and #RADIOs (shared variance)? (Provide an actual value.)

5. What is total variation in LIFEEXP explained by #DOCs and #RADIOs? (Provide an actual value.)

6. What is the value of Multiple R? (Provide an actual value.)

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Solution: The downloadable solution consists of 14 pages, 1459 words and 12 charts.
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