INSTRUCTIONS: Answer all questions in the spaces provided (or indicate clearly where you have continued your answer). Calculators are NOT permitted. Reduce all computations to the simplest form so that anyone with a calculator could attain the answer easily. Show your work and reasoning to the fullest extent possible so that part marks can be assigned as warranted. You have 75 minutes to complete this exam. All parts of both questions are worth 10 points (and some are much easier than others). Total points = 150. This means roughly 5 minutes for each answer. Budget your time carefully. NOTE: these data are fictitious.
 

SCENARIO: Your consulting firm has been hired by the Department of Environmental Health and Safety. The DEHS would like you to analyze the relationship between the annual number of toxic releases (per 100 establishments) for dry-cleaning firms (spills_i) and some policy variables deemed relevant for the control of these accidents. You have been provided with a sample of data from 19 randomly selected jurisdictions. The variables they have given you include the average age of such establishments in the jurisdiction (age_i), the number of full-time inspectors per 100 establishments in the jurisdiction (insp_i), the average legal penalties imposed for toxic releases in that jurisdiction (in thousands of dollars) (pen_i), and the median income of households in the jurisdiction (in thousands of dollars) (medinc_i). The statistical analyses you perform are given in the Exhibits.
 

1. Begin gently. Fill in the blanks:

Across these 19 jurisdictions, what is the mean "number of toxic releases per 100 establishments"? ______

What is the highest observed "number of inspectors per 100 establishments"? ________

What is the standard deviation in "average legal penalties per release" across the sample? ________

Do the descriptive statistics you have just provided refer to the joint distribution of these three variables, or to their marginal distributions? ______________

What is the correlation between insp_i and pen_i in this sample? ________

What are the units for this correlation measure? ________
 

2. Using the descriptive statistics only, test the hypothesis that the true marginal mean "number of toxic releases per 100 establishments" is 4 per year.
 

3. Does Regression 1 make sense? Why or why not?
 

4. The Administrator for the DEHS says "If the number of inspectors in a jurisdiction has no statistically discernible effect on the number of toxic releases from these establishments, why are we paying the salaries of these people?" Based upon the relevant simple regression in the Exhibits, is it possible that there is a downward-sloping relationship between the number of inspectors and the number of releases? Explain how you have reached this conclusion.
 

5. Based on Regression 3, test the hypothesis that in order to reduce the "number of releases per 100 establishments", on average, by one per year, it would be necessary to increase the average legal penalty per spill by $5,000.
 

6. Based on Regression 3, what average number of releases per 100 establishments would you expect for a jurisdiction with an average legal penalty of $50,000 per release? Give the precise formula for a point estimate and explain explicitly how a 95% confidence interval for this prediction would be constructed. Why should you use caution in making the this prediction?
 

7. You think for a while and then realize that the number of toxic releases per 100 establishments is probably a joint function of several different factors, rather than just one at a time. You estimate Regression 6 in order to ascertain the joint effects of all available determinants on the average number of toxic releases per 100 establishments. Describe what seems to happen to the apparent effect of the insp_i variable when you include the other variables in your model. If this apparent effect is different, explain why. What do you tell the Administrator of the DEHS?
 

8. In Regression 6, explain the use of the / auxrsqr option on the ols command. What does it tell you here?
 

9. For Regression 6, test the hypothesis that the age_i variable does not belong in the model. What do you conclude?
 

10. One employee of the DEHS, who has worked there for decades, claims that higher expected penalties for infractions can work as a substitute for greater monitoring of establishments by inspectors. In fact, she says, if you can work to write higher penalties into the regulations, a $1000 higher expected penalty for violations is as good at preventing spills from happening as the presence of one more full-time inspector. Test this hypothesis statistically.
 

11. In the different specifications in the Exhibits, what proportion of the variation in toxic releases across jurisdictions can be explained by a model that uses only the average legal penalty? _______ What proportion can be explained by a model that uses the average legal penalty, the number of inspectors, the average ages of facilities and median incomes? ______ Can these be compared? Why or why not?

 
12. The estimated model in Regression 6 exhibits a positive intercept. What is the interpretation of this point estimate? Does it make sense to test hypotheses about the size of the coefficient on the constant term? Explain.

 
13. In Regression 6, are the slope coefficient on the age_i and medinc_i variables individually statistically significantly different from zero? _____ Does our inability to discern the separate effect of age_i and medinc_i on spills_i stem from multicollinearity problems? Explain.

 
14. Is the model in Regression 7 an adequate model for toxic releases by these establishments? Explain your reasoning. Can we conclude that there are no other relevant determinants of the average annual number of releases for this set of jurisdictions? Explain.

 
15. (i.) Specifically, what do we call the distribution that appears in the histogram at the end of the Exhibits?

(ii.) Specifically, what do we call the scatterplot that appears at the end of the Exhibits?