Econ 57 Fall 2002 Final Examination (exactly 150 minutes)
Answer all 20 questions, leaving tedious calculations undone.

1. Consumer Sports tested the gasoline mileage of three subcompact cars by driving 10 of each model from Claremont, California, to Brewster, Massachusetts. Do not do an ANOVA F test, but do explain why you believe that such a test either would or would not have a P value less than 0.05. What is the null hypothesis?

 model A 32 31 30 27 31 33 31 34 28 31 model B 36 41 43 41 41 42 37 40 39 42 model C 39 37 33 35 32 36 36 34 37 33

2. Dr. Timothy Roberts told a 2002 meeting of the Pediatric Academic Societies in Baltimore that a 1995-1996 U.S. government survey of 4,600 high-school students found that both boys and girls were much more likely to have smoked, used alcohol and marijuana, had sex, and skipped school if they had body piercings. Does his study imply that we could reduce these activities by teenagers by making body piercings illegal?

3. A student reported taking two random samples of size 600 (with replacement)

a. from a population of 1000 containing 400 women and 600 men, and obtaining 247 women

b. from a population of 10,000 containing 4000 women and 6000 men, and obtaining 248 women

If these were random samples, what is the probability that the difference between the number of women in the two samples would be less than or equal to 1? Use a normal approximation.

4. You are playing draw poker and have been dealt a pair of fours, a jack, a seven, and a three. You keep the pair of fours and discard the other three cards. What is the probability that at least one of the three new cards you receive will be a four?

5. Fifty six college students participated in an ESP test in which a professor flipped a coin ten times and each student attempted to identify each flip as a head or tail. If a student’s guess is equally likely to be a head or tail, regardless of his or her previous guesses, what is the probability that a student will end up with 10 guesses consisting of exactly 5 heads and 5 tails?

6. Let p be the probability you found in the preceding exercise. It turned out that 27 of these 56 students ended up with 10 guesses consisting of exactly 5 heads and 5 tails. Use these data to test the null hypothesis that a student’s guess is equally likely to be a head or tail, regardless of his or her previous guesses.

7. A carnival game involves two boxes, a small upper box and a large lower box. If the contestant aims for the upper box, there is a 0.5 probability the ball will land in this box and a 0.5 probability it will land in the box below. If the contestant aims for the lower box, it will always land in the lower box. The contestant tosses two balls, one after the other, and wins if exactly one ball lands in each box. What is the best strategy? What is the probability of winning?

8. An economics professor told his students that instead of spending hundreds of dollars for a very accurate watch, they should wear 20 cheap watches and calculate the average time. Suppose that the reported time on each cheap watch is normally distributed with a mean equal to the actual time and a standard deviation of 10 seconds, and that the errors in the reported times on the cheap watches are all independent of each other. What is the value of the mean and standard deviation of the average time on these 20 watches?

9. The Mandarin, Cantonese, and Japanese pronunciation of “four” and “death” are almost identical, and many Chinese and Japanese people consider the number 4 to be unlucky. Californians have some say in the last four digits of their telephone numbers. A study of the last four digits of the telephone numbers of 1984 Chinese and Japanese restaurants in California counted the number of times the digits 0 - 9 appeared:

 digit 0 1 2 3 4 5 6 7 8 9 occurrences 960 834 789 725 562 650 726 657 1332 701
Show how you would test the null hypothesis that all 10 digits are equally likely.

10. A sociology professor argued that the stress associated with the number of 4 might cause Chinese and Japanese persons to suffer an unusually large number of fatal heart attacks on the fourth day of the month. Mortality data for Chinese and Japanese Californians for the years 1989-1998 show that 472 of the 1391 fatal heart attacks that occurred on days 3, 4, or 5 of a month occurred on day 4. Use these data to test the null hypothesis that of those fatal heart attacks occurring on days 3, 4, or 5 of a month, there is a 1/3 probability that the death will be on day 4. (Use a normal approximation.)

11. A taxi hit a pedestrian one night and fled the scene. The entire case rests on the testimony of a man who saw the accident from his window. He says that he saw the pedestrian struck by a blue taxi. In trying to establish her case, the lawyer for the injured pedestrian establishes the following facts:

1. There are only two taxi companies in town, Blue Cabs and Black Cabs. On the night in question, 85% of the taxis on the road were black and 15% were blue.

2. The witness underwent an extensive vision test under conditions similar to those on the night in question, and demonstrated that he can successfully distinguish a blue taxi from a black taxi 80% of the time.

Based on this information, what is the probability that the pedestrian was hit by a blue taxi?

12. One hundred and twenty college students who had been in a serious romantic relationship in the past two years were asked to name the month in which their most recent relationship began. The chi-square value was 18.6 and, with 11 degrees of freedom, the p-value was 0.0686. Explain why you either agree or disagree with this interpretation of the results: “This study concludes that the data are statistically substantial because there are more than 5 degrees of freedom, and statistically significant because the p value is not below 0.05.”

13. This multiple regression equation was estimated: Y = -.7515 + 0.47F + 0.507M + 0.047A + 4.33D, where y = height of student, F = height of student’s biological father, M = height of student’s biological mother, A = student’s age, and D = 1 if the student is male, 0 if female. Explain why you either agree or disagree with this interpretation of the results: “Males were, on average, 4.33 inches taller than females.”

14.In the gambling game Chuck-A-Luck, a player can bet \$1 on any number from 1 to 6. Three dice are thrown and the payoff depends on the number of times the selected number appears. (For example, if you pick the number 2, your payoff is \$4 if all three dice have the number 2.) What is the expected value of the payoff?

 Number of dice with number 0 1 2 3 Payoff (dollars) 0 2 3 4

15. Marcus Lee suggested a betting strategy based on the logic of regression to the mean and data on how professional football teams had done in recent weeks. Do you think that he recommended betting for or against teams that had been doing poorly? Explain clearly and concretely how the logic of regression to the mean applies to football games.

16. A study found that with simple regression equations, the length gain during a human baby’s first three months of life is negatively correlated with the baby’s birth length and also negatively correlated with the baby’s birth weight; however, a multiple regression equation showed the length gain to be related negatively to birth length and positively to birth weight. Explain these seemingly contradictory findings.

17. A study that compared the grades given to male and female essays said that one flaw in the study was that “the number of male essays graded was not equal to the number of female essays graded.” Explain why this either is or is not a flaw.

18. A study comparing the grades given by female and male students to an essay written by a male found that female graders gave an average grade of 7.125 and male graders gave an average grade of 7.500. Explain why you either agree or disagree with this interpretation of the results: “The t-value was 0.5800, with a p-value of 0.5721. This can be interpreted as there being a 57.21% chance that the means of 7.500 for male graders and 7.125 for female graders were 0.5800 standard deviations away from each other.”

19. A student estimated a regression model that used the number of pages in best-selling hardcover books to predict their prices. Explain why this conclusion is wrong: “The 0.75 R-squared is pretty large, when you consider that the price of a book is \$20; i.e., \$15 of the price is explained by the number of pages.”

20. Explain this report <http://stufs.wlu.edu/~hodgsona/bingodeaths.htm>:

A recent study compared the death rates of people who play a variety of games including: Bingo (41.3), Beer Pong (1.6), and Mortal Combat (0.04). The numbers reveal an alarming truth: bingo may not be as safe as some people have assumed.

“This is the first evidence we have seen that violent video games actually reduce the death rate,” says PlayGear CEO Pete Elor, “It comes as a blow to the head for people who advocate less violent forms of entertainment.” Lawyer Gerald Hill thinks the church and community need to take action: “When you look at the numbers, there’s just no way you can get around it. Bingo is claiming lives right and left. And it doesn’t just affect the Bingo community. Friends and families of those we lose to Bingo suffer even a greater loss.” We can only hope that this study will cause people to think twice before engaging in such risky behavior.

-Ashley Hodgson, special reporter