Economics 57: Fall 2000 Midterm Answers

Econ 57 Fall 2000 Midterm Answers

1. The number of pages haven't been adjusted for the size of the yellow pages and hence the size of the cities. (These same researchers found that personal bankruptcy rates tend to be higher in large cities than in smaller ones.) Also, the selection of 7 cities for one group and 5 for the other sounds like data mining. Perhaps most importantly, the causation may run the other way around: cities with lots of personal bankruptcy filings tend to encourage bankruptcy lawyers to set up practices.

2. (2/3)⁶³ = 8.06 * 10^-12, or 1 in 124 billion.

3. The binomial probability is

4. The larger the sample size, the more likely it is that the sample success proportion x/n will be close to the success probability, 0.2. Therefore, with 20 questions, the sample success proportion is less likely to be greater than 0.5. 5. If a person has the disease, the probability of 2 positive readings is 0.9(0.9) = 0.81; if a person does not have the disease, the probability of 2 positive readings is 0.1(0.1) = 0.01. Out of every 100,000 people, we can expect 100 to have the disease and 0.81(100) = 81 of these people to test positive; we can expect 99,900 to not have the disease and 0.01(99,900) = 990 to test positive. Therefore, among those with two positive readings, the probability of disease is 81/(81 + 999) = 0.075. Here is a complete contingency table:

	2 positive	1 positive, 1 negative	2 negative	Total
Has Disease	81	18	1	100
No Disease	999	17,982	80,919	99,900
Total	1080	18,000	80,920	100,000

We can also answer this question by using Bayes Rule.

6. E[profit] = ($1000 - $10,000) P + $1,000(1 - P) = $1,000 - $10,000 implies P > 0 if P < 0.10.

7. The z values are

The respective probabilities are P[z > 2.49] = 0.0064 and P[z > 3.24] = 0.0006.

8. Think of each slot as numbered 1, 2, ..., 9:

The number of possible combinations is 9(8)(7)/(3)(2)(1) = 84, of which 8 are winning combinations: 123, 456, 789, 147, 258, 369, 159, 357. Therefore, the probability of winning is 8/84 = 0.095

9. There is clearly selection bias; for example, those whose first language isn't English are less likely to read this newspaper.

10. For a histogram, we should divide the 100 annual data into a small number of categories, such as 0-10 inches, 10-20 inches, and so on. The density is the fraction of the total number of years with rainfall in that interval, divided by the interval width.