Econ 57 Fall 2000 Final Examination (exactly 75 minutes)
Answer all 10 questions, leaving tedious calculations undone.

1. Legendary Harvard University statistics professor Frederick Mosteller reported that if you flip a coin in class and ask if anything suspicious is happening, "hands suddenly go up all over the room" after the fifth head or tail in a row, "so there is some empirical evidence that the rarity of events in the neighborhood of .05 begins to set people's teeth on edge." What is the probability that a fairly flipped coin will give either five heads or five tails in a row?

2. To see who serves first in their Thursday squash games, Player A spins the racket while Player B guesses whether the racket logo will stop face up or face down. Player B initially believes that there is only a 5% chance that A cheats when he spins the racket. But after A wins the first five times he spins the racket, B isn't so sure. Assuming that A will always win if he cheats and has a 50% chance of winning if he doesn't cheat, what is B's revised probability that A cheats?

3. The September/October 2000 Claremont College's Commuter Chronicle reported that, "As of early August, two-thirds, or 8 out of the 12, ozone Health Advisories this year had occurred on weekends." Are these data statistically persuasive?

4. Least squares will be used to estimate the model Y = a + bX + e, using annual data for 1991 through 2000. If X = 100 and Y = 100 in 1991 and X = 200 and Y = 200 in 2000, can the least squares estimate of b possibly be negative? Explain your reasoning.

5. A French hospital put 412 patients who had suffered one heart attack on a traditional Mediterranean diet (including olive oil, fruit, and bread); the control group consisted of 358 heart-attack patients who were given a recommended low-fat diet. Four of the patients on the Mediterranean diet and 17 of the patients on the low-fat diet suffered a second heart attack during the two years of the study. Is this observed difference substantial and statistically persuasive?

6. A test of the null hypothesis that the average Pomona College student gains 15 pounds during the first year at college surveyed 100 students and obtained a sample mean of 4.82 pounds with a standard deviation of 5.96 pounds. Explain why you either agree or disagree with each of these conclusions:

a. "We assumed that the standard deviation of our sample equaled the population standard deviation."

b. "We calculated the standardized z value to be z = -17.08. The two-sided p value is 2.82 x 10-31. According to Fisher's rule of thumb, our data are not statistically significant because our p value is considerably less than 0.05. This indicates that the probability of observing a large difference between the sample mean and the population mean that the null hypothesis predicts is greater than 0.05."

c. "Our data strongly indicate that the Freshman 15 is just a myth. However, it must be recognized that we only took one sample of 100 students. Perhaps if we took other samples, our results would be different."

7. An ANOVA test obtained an F value of 0.0; what is the P value?

8. A study of injuries suffered by Pomona varsity athletes during the years 1980-1995 obtained these data:

  Females Males
Basketball 214 344
Soccer 155 236
Swimming 161 34
Tennis 82 82
Track 151 233
     Identify the patterns in these data and determine whether they are statistically persuasive.

9. Five polls completed on November 5, 2000, gave these results:

  number surveyed Bush (%) Gore (%) Margin of Error
CNN/USA Today/Gallup 2386 47 45 2
IBD/CSM/TIPP 989 48 42 3
ICR 1000 46 44 3
Reuters/MSNBC 1200 47 46 3
VOTER.COM 1000 46 37 3
     Combine these polls to estimate the percentage of the vote that Bush would have received if the election had been held that day. Give a 95% confidence interval for this prediction. (Remember that you can just set it up without doing any actual calculations.)

10. Answer this question to Ask Marilyn:

I recently returned from a trip to China, where the government is so concerned about population growth that it has instituted strict laws about family size. In the cities, a couple is permitted to have only one child. In the countryside, where sons traditionally have been valued, if the first child is a son, the couple may have no more children. But if the first child is a daughter, the couple may have another child. Regardless of the sex of the second child, no more are permitted. How will this policy affect the mix of males and females?