**1.** Self-selection bias. The self-esteem of children who choose to participate may differ systematically from the self-esteem of nonparticipants. If so, we won't know whether the observed differences are due to the participation or to the kinds of children that participate. We might take a group of children who sign up for soccer and allow only a randomly selected subset to participate, leaving the rest as a control group. (It's cruel, but done all the time in medical research.)

**2.** Survivor bias. Companies that are now big may have gotten big by growing fast! The study should have looked at companies that were big 20 years ago and compared their growth rates over the succeeding 20 years to the growth rates of other companies.

**3.** The black athletes on television are hardly a random sample and are therefore unrepresentative of "blacks as a group."

**4.** Using a contingency table with 100 people brought to trial, of which 90 (90 percent) are guilty. Letting 90 percent of the 10 innocent defendants be set free and 90 percent of the 90 guilty defendants be convicted, the complete table is

Convicted | Set Free | Total | |

Innocent | 1 | 9 | 10 |

Guilty | 81 | 9 | 90 |

Total | 82 | 18 | 100 |

Thus, the answers to the questions asked are:

a. Of those people convicted, 1/82 = 0.012 are innocent.

b. Of those people set free, 9/18 = 0.50 are guilty.

**5.** It doesn't matter. Imagine that each person drew a slip of paper without looking at it. Would it matter who looked first?:

**6.** Each die gives you a 1/6 chance of getting $1, an expected payoff of $1/6. With 3 independent dice rolls, the expected payoff is 3($1/6) = $1/2. We also need to consider that the player gets the original dollar back if any die is a winner. The probability of doing so is 1 minus the probability of not doing so: 1 - (5/6)3 = 91/216 = 0.421. Thus the player, on average gets $0.921 back on a $1 bet.

Here's a more straightforward way. This is a binomial problem with n = 3 and p = 1/6. The probabilities of 0, 1, 2, and 3 wins are:

The expected value of the net profit is 0.5787(-1) + 0.3472(1) + 0.0694(2) + 0.0046(3) = -0.079

**7.** The SOI is the z value multiplied by 10. A SOI reading of -22.8 corresponds to a z value of -2.28, which has a probability of 0.0113.

**8.** The probability that Smith will need enhancement surgery on at least one eye is equal to one minus the probability that Smith will not need enhancement surgery on either eye: 1 - (0.7)(0.7) = 0.51.

**9.** We need to know the fraction of all marriages that occurred in each month. In fact, because there are relatively more marriages in June and August (11.84 percent of all marriages take place in June and 11.70 percent in August), an examination of most marital characteristics will tend to find a preponderance of June and August marriages.

**10.** The omission of zero from the vertical axis magnifies the index's zigs and zags. Between 2:00 and 2:30, for example, the height of the line drops by 80 percent; yet, the actual decline in the Dow was only about sixth-tenths of one percent.