Econ 57 Spring 1997 Final Exam

1. Explain why the following visual display of household income data in Sierra Madre, California, is misleading and then redraw the figure correctly.

2. There are 12 astrological signs (Aquarius, Scorpio, and so on.) If a person's birthday is equally likely to be in each of these 12 signs, what is the probability that in a group of 4 persons, at least 2 will have the same sign?

3. Each semester, Professor Smith gives students a copy of the final exam from the semester before to help them study for their final. Once, as he passed out the previous semester's exam, he warned, "There is inevitably variation in the difficulty of tests. This exam was one standard deviation below the mean." Assuming normality, how would you interpret this remark?

Say 10 tickets are numbered 1 through 10 in a drawing. Half the numbers are even and half are odd. The first ticket is drawn, and it's No. 3 which is odd. That leaves five even numbers and four odd ones. Doesn't this mean that the next ticket to be drawn is more likely to be even? If I buy a ticket at this point, wouldn't I have a better chance of winning the next draw by choosing an even number?

5. Use regression toward the mean to explain why a movie sequel is usually not as good as the original.

6. Suppose that the damage award favored by individual potential jurors in a particular personal injury case can be described by a normal distribution with a mean of \$5,000,000 and a standard deviation of \$2,000,000. (This probability distribution is across randomly selected jurors.) What percentage of potential jurors favor an award of more than \$5,500,000? Of less than \$4,500,000?

7. Continuing with the preceding exercise, assume that a jury is a random sample from this distribution and that the jury's damage award is the average of the awards favored by those who serve on the jury. Carefully explain the differences in rewards that can be anticipated with a 6-person jury system versus a 12-person system.

8. Without doing any calculations, explain why you believe the two-sided P value for a test of H0: m1 = m2 to be either larger than 0.05 or smaller than 0.05 for these data:

 Sample 1 Sample 2 4.0 4.8 3.4 3.5 4.1 7.2 6.7 5.8 6.5 6.7 4.6 4.0 4.4 3.9 4.3 6.7 5.9 5.8 5.8 6.0 4.8 3.4 4.6 4.0 4.4 5.8 6.0 6.7 5.8 6.2 4.8 3.4 4.0 3.5 4.1 6.5 6.7 7.2 6.7 5.8 3.9 4.3 4.6 4.0 4.4 6.0 5.8 6.7 5.9 5.8 3.4 4.6 4.8 4.0 4.4 5.8 6.0 6.2 5.8 6.7 3.4 4.0 4.8 3.5 4.1 7.2 6.5 6.7 6.7 5.8 4.0 4.4 3.9 4.3 4.6

9. Use these data on the number of households (in millions) of different sizes to draw two histograms, one for 1890 and one for 1990. (Calculate the relative frequency for the 7-or-more category, but do not include this category in your histogram.) What important differences do you detect in these two histograms?

 Size 1 2 3 4 5 6 7 or more 1890 0.457 1.675 2.119 2.132 1.916 1.472 2.919 1990 23 30.2 16.1 14.6 6.2 2.2 1.5

10. The preceding exercise shows the number of households of different sizes in 1990.

a. Looking at the 93.8 million households, ordered by size, what is the size of the median household?

b. Looking at all 264 million individuals, ordered by the size of the household they live in, what is the size of the household that the median individual lives in?

c. Pomona College reports its average class size as 14 students per class. However, a survey that asked Pomona College students to give the size of the classes they were enrolled in found that the average class size was 38 students and that 70 percent of the respondents' classes had more than 14 students. Use the insights gained in the first two parts of this exercise to explain this disparity.

11. In the United States, 36 of 50 states have a death penalty. To examine the relationship between the death penalty and murder rates, the following equation was estimated by least squares using 1987 data:

where y = murder rate (murders per 100,000 persons); x = 1 if state has a death penalty, 0 if it doesn't; and the standard errors are in parentheses.

a. What does R-squared mean and how is it measured?

b. Is the relationship statistically significant at the 5 percent level?

c. What is the average value of the murder rate?

d. The researcher added the District of Columbia to his data and obtained these results:

The murder rate in D.C. is 36.2; the state with the highest murder rate is Michigan, with 12.2 murders per 100,000 persons. Explain how a comparison of the two estimated equations persuades you that D.C. either does or does not have a death penalty.

e. Explain the error in this interpretation of the results including D.C.: “Those of us who oppose the death penalty can breathe a sigh of relief, now armed with statistically significant evidence that the death penalty is only cruel and not effective.”

12. In a 1982 racial-discrimination lawsuit, the court accepted the defendant's argument that racial differences in hiring and promotion should be separated into eight job categories. In hiring, it turned out that blacks were underrepresented by statistically significant amounts (at the 5 percent level) in four of the eight job categories. In the other four categories, whites were underrepresented in two cases and blacks were underrepresented in two cases, though the differences were not statistically significant at the 5 percent level. The court concluded that four of eight categories was not sufficient to establish a prima facie case of racial discrimination. Assume that the data from these eight job categories are independent random samples.

a. What is the null hypothesis?

b. If the null hypothesis is true and the court requires statistical significance at the 5 percent level in all eight job categories, what is the probability that the null hypothesis will be rejected?

c. If the null hypothesis is true and the court requires statistical significance at the 5 percent level in at least one of the eight job categories, what is the probability that the null hypothesis will be rejected?

13. Continuing with the preceding exercise:

a. Explain why data that are divided into eight job categories might not show statistical significance in any of these job categories, even though there is a statistically significant relationship when the data are not disaggregated.

b. Explain why data that are divided into eight job categories might show statistical significance in each of the eight categories, even though there is not a statistically significant relationship when the data are not disaggregated.

14. A researcher used data on these three variables for 60 universities randomly selected from the 229 national universities in U.S. News & World Report's 1995 rankings of U.S. colleges and universities: y = graduation rate, mean 59.65; x1 = student body's median SAT, mean 1,030.5; and x2 = percent of student body with GPAs among the top 10 percent at their high school, mean = 46.6. Here are the results:

Interpret these results. Be sure to explain why the estimated coefficients of x1 and x2 are lower in the third equation than in the first two.

15. Explain the error in this critique of a statistics paper in which the z-value turned out to be 2.8: "Even though the z-value was larger than 2, the sample size was only 24. The results could be invalidated by sampling error due to the small sample."

16. If six standard six-sided dice are rolled simultaneously, what is the probability that all six numbers will appear, not necessarily in order?

17. Here are the predicted and actual daily high and low temperatures at the Los Angeles Civic Center for every fourth day during the period from November 7, 1996 to February 15, 1997:

 Sample 1 Sample 2 Daily Low Daily High Predicted Actual Error Predicted Actual Error 11/7/96 50 51 -1 81 73 8 11/11/96 60 63 -3 87 91 -4 11/15/96 58 56 2 73 62 11 11/19/96 56 54 2 71 71 0 11/23/96 54 59 -5 67 67 0 11/27/96 58 54 4 83 80 3 12/1/96 46 46 0 67 66 1 12/5/96 51 50 1 70 59 11 12/9/96 55 53 2 67 57 10 12/13/96 52 56 -4 72 68 4 12/18/96 48 50 -2 71 77 -6 12/21/96 47 50 -3 68 59 9 12/25/96 52 50 2 70 75 -5 12/29/96 52 53 -1 62 66 -4 1/2/97 56 60 -4 67 64 3 1/6/97 47 52 -5 60 62 -2 1/10/97 48 47 1 70 62 8 1/14/97 48 47 1 56 57 -1 1/18/97 47 51 -4 74 79 -5 1/22/97 49 51 -2 62 56 6 1/26/97 55 54 1 60 58 2 1/30/97 58 56 2 83 78 5 2/3/97 49 55 -6 67 66 1 2/7/97 51 49 2 66 70 -4 2/11/97 51 50 1 64 62 2 2/15/97 50 49 1 75 77 -2

Test the null hypothesis that the probability that the daily low temperature will be at least 50 degrees Fahrenheit is the same in each of these months

18. Use two box plots to display the data in Exercise 17 on the actual low and high temperatures.

19. Use the data in Exercise 17 to determine a 95 percent confidence interval for the population mean error predicting the daily low temperature and to test at the 5 percent level the null hypothesis that the population mean error predicting the daily low temperature is zero.

20. In March of 1992, The Wall Street Journal reported that, "Foreign stocks and foreign-stock mutual funds have been miserable performers since early 1989, which suggests a rebound is long overdue." The article went on to quote the chief investment officer of the G.T. Global mutual funds group: "The U.S. stock market has outperformed the EAFE index [of European, Australian, and Far East stocks] for three consecutive years. If the U.S. were to outperform the EAFE index in 1992, it would be the first [such] four-year stretch in the 20 years that the EAFE index has been compiled. If history is any guide, it's about time that foreign markets started doing better." Explain the error in their statistical reasoning.