Econ 57 Spring 1999 Midterm

1. Identify the most important statistical problem with this study and explain how you would redo the study to get rid of this problem: A study of the effects of youth soccer programs on self-esteem found that children who had played youth soccer had more self-esteem than children who had not participated in youth soccer programs.

2. Identify the most important statistical problem with this study and explain how you would redo the study to get rid of this problem: A study of the 30 largest U.S. companies found that their average growth rate over the preceding 20 years had been well above the average growth rate for all companies, suggesting that big companies grow faster than the average company.

3. In 1993 Dr. George Graham, a member of a presidential Task Force on Food Assistance, stated that, "If you think that blacks as a group are undernourished, look around at the black athletes on television--they're a pretty hefty bunch." ["Big Changes Made in Hunger Report," San Francisco Chronicle, September 6, 1994] Identify the most important statistical problem with this conclusion.

4. In the United States, criminal defendants are presumed innocent until they are proven guilty beyond a reasonable doubt, because it is thought "better to let nine guilty people go free than to send one innocent person to prison." Assume that 90 percent of all defendants are guilty, 90 percent of the guilty defendants are convicted, and 90 percent of the innocent defendants are set free. Of those people convicted, what percent are innocent? Of those people set free, what percent are guilty?

You're at a party with 199 other guests when robbers break in and announce that they are going to rob one of you. They put 199 blank pieces of paper in a hat, plus one marked 'you lose.' Each guest must draw, and the person who draws 'you lose' will get robbed. The robbers offer you the option of drawing first, last or at any time in between. When would you take your turn? [Marilyn Vos Savant, "Ask Marilyn," Parade Magazine, January 3, 1999]

At a monthly 'casino night,' there is a game called Chuck-a-Luck: Three dice are rolled in a wire cage. You place a bet on any number from 1 to 6. If any one of these three dice comes up with your number, you win the amount of your bet. (You also get your original stake back.) If more than one die come up with your number, you win the amount of your bet for each match. For example, if you had a \$1 bet on the number 5, and each of the three dice came up with 5, you would win \$3. In the long run, what is your average profit on each \$1 bet? [Marilyn Vos Savant, "Ask Marilyn," Parade Magazine, December 27, 1998]

7. The Australian Bureau of Meteorology uses the monthly air pressure difference between Tahiti and Darwin, Australia, to calculate the Southern Oscillation Index: SOI = 10(x - m)/s, where x is the air-pressure difference in the current month, m is this particular month's historical average air-pressure difference, and s is the standard deviation of this month's historical air-pressure difference. Negative values of the SOI indicate an El Nišo episode, which is usually accompanied by less-than-usual rainfall over eastern and northern Australia; positive values of the SOI indicate a La Niša episode, which is usually accompanied by more-than-usual rainfall over eastern and northern Australia. If x is normally distributed with a mean of m and a standard deviation of s, what is the probability of an SOI reading as low as -22.8, which occurred in 1994?

8. Smith had Lasik surgery to correct astigmatism in both his eyes. Before the surgery, his doctor told him that, because of the severity of the astigmatism, for each eye there was a 30% chance that a second operation would be required to enhance his vision. Assuming that the results are independent, what is the probability that he will need enhancement surgery on at least one eye?

9. The Public Health Service has tabulated data on the number of divorces, according to the month in which the marriage ceremony was performed. Thus 6.4 percent of all divorces involve couples who were married in January. Do these data show that persons who are married in June or August are more likely to get divorced than are people married in January?

 Month Married Fraction of Divorces Month Married Fraction of Divorces January 0.064 July 0.087 February 0.068 August 0.103 March 0.067 September 0.090 April 0.073 October 0.078 May 0.080 November 0.079 June 0.117 December 0.087

10. Explain why this graph of the Dow Jones Industrial Average of stock prices on January 27, 1997 gives a misleading visual impression of the volatility of stock prices that day:

[Adapted from a graph in the Business section of the Los Angeles Times, January 28, 1997]