Answer all 10 questions; you have exactly 75 minutes.
ALWAYS explain your reasoning
1. Researchers counted the number of pages of bankruptcy-lawyer ads in the Yellow Pages of 7 cities that had a large per capita rate of personal bankruptcy filings and in 5 cities that had low rates. There were an average of 9.6 pages in the high-bankruptcy cities and only 3.8 pages in the low-bankruptcy cities, suggesting that bankruptcy-lawyer ads are responsible for the variations in the number of personal bankruptcy filings. Why should we interpret these results cautiously?
2. In 2000, Sandbox.com offered a $10 million prize to anyone who could correctly pick the winners of all 63 NCAA Tournament basketball games. If, in each game, one team has a 2/3 chance of winning and the other team has a 1/3 chance, and you always pick the team with the better chance, what is your probability of winning the $10 million prize?
3. William is running out of time on a multiple-choice exam. There are 20 questions left, each with 5 possible answers. He quickly marks down 20 answers, without even reading the questions. What is the probability that he will get more than half of these questions correct?
4. (continuation) Is William more likely to get (a) more than 5 questions right if he guesses the answers to 10 questions, or (b) more than 10 questions right if he guesses the answers to 20 questions? You must use logic, not calculations, to answer this question.
5. One out of 1000 people have a certain disease. If a person has this disease, a test will give a positive result 90% of the time; if a person does not have the disease, the test will give a negative result 90% of the time. If a person tests positive in two independent tests, what is the probability that this person has the disease?
6. Defendants awaiting trial are often allowed to leave jail if they leave a cash amount (a "bond") that is forfeited if they do not return for their trial. Bail bondsmen put up the requisite cash in return for a payment from the defendant. For example, a bail bondsman might put up a $10,000 bond after the defendant pays the bondsman $1,000, which the bondsman keeps whether or not the defendant returns for trial. In this example, for what values of P, the probability that the defendant will disappear (causing the bondsman to lose the $10,000 bond), is the expected value of the bondsman's profit greater than 0?
7. In his 1868 work, Carl Wunderlich concluded that temperatures above 100.4 degrees Fahrenheit should be considered feverish. In a 1992 study, Maryland researchers suggested that 99.9 degrees Fahrenheit was a more appropriate cutoff. If the oral temperatures of healthy humans are normally distributed with a mean of 98.23 and a standard deviation of 0.67, what fraction of these readings are above 100.4? Above 99.9?
8. In one carnival game, three balls are tossed into a tic-tac-toe arrangement of nine slots. The slots are surrounded by a fence, so that each ball lands in a slot; only one ball fits in each slot. If the balls land randomly, what are the chances of a tic-tac-toe (three balls in a row horizontally, vertically, or diagonally?
9. On March 4, 1996, a major California newspaper, The Sun, asked its readers the following question: "Do you think the English-only law should be enforced in California?" Of the 2,674 persons who mailed in responses, 94 percent said yes. Why might this newspaper's readers be a very biased sample of the opinions of California residents?
10. A researcher recorded annual precipitation in Rhode Island for 100 years, 1898-1987. What is wrong with this histogram? (Density is the rainfall for each year divided by the total rainfall for all 100 years.)