Economics 57 Spring 1997 Midterm Answers

1. a. 6 (four 6s)
b. 1 (four 1s)
c. 2.89 (two 1s and two 6s gives a mean of 3.5 and a variance of 4(2.52)/(3 - 1) = 8.333)
d. 0 (the same number on each roll)

2. Using the multiplication rule, a guesser has a 1/3 probability of identifying correctly the beer from a tap. Given that this guess is correct, there is a 1/2 probability of identifying correctly the beer from a can. If the first two guesses are correct, the third must be too. The probability of identifying all three correctly is consequently (1/3)(1/2)(1) = 1/6. If many beer experts were never correct, this suggests (contrary to the advertisement) that the three beers do taste different, but not in the way that the beer experts think they will taste different.

3. We can use a contingency table as in the textbook, assuming that a population of 10,000 people are tested:

Test +
Test -

Thus P[drug-free | positive reading] = 475/950 = 0.50

Or we can use Bayes' theorem:

4. This question can be answered with either the binomial distribution or with a normal approximation to the binomial distribution. For the binomial distribution, pi = 0.5, n = 2000, and

A normal approximation with continuity correction is

5. Each contestant has the same chance of winning. The first contestant can win in three ways: by choosing the right box on the first; by choosing wrong, the other person choosing wrong, and then making the right choice; or by four bad choices followed by the right choice. The probability that the first contestant will win the prize is

6. a. 10 billion times
b. 10 billion times
c. 10 billion times
d. 100 billion times
e. 100 billion times

7. Because the origin is at 230,000, rather than 0, a visual impression of great customer growth is created. The height of the water in the 1994 graph is four times the height of the water in the 1990 graph. Yet, the number of customers increased by only 6.4 percent: (250 - 235)/235 = 0.064.

8. If the responses ranged from 0 to 30 with a mean and median below 5, the mean should have been pulled above the median by the 30 response (and any other high answers). The graph, however, is drawn with the mean below the median. The legends are reversed, in that the diamond is the median and the circle is the mean.

9. Here is a plausible histogram, reflecting the guesstimates that 30 percent are 20-40 years old, 30 percent are 40-50, 30 percent are 50-60, and 10 percent are 60-80:

10. The exact answers are less important than whether you use a reasonable procedure. The histogram seems to be roughly bell-shaped with a center at slightly less than 40 inches. Because the histogram is reasonably symmetrical, we might use 40 inches for the mean. For the standard deviation, we can use the rule of thumb that for a bell-shaped histogram, approximately 95 percent of the data are within two standard deviations. It appears that roughly that 95 percent of these data are between 35 and 45. If two standard deviations is equal to 5, then the standard deviation is 2.5 inches.