HOMEWORK #5 ANSWERS

Chapter 20

6. Option (ii) is right. In absolute terms, the California sample will be much bigger, and therefore more accurate for estimating percents; the size of the sample relative to the size of the state is basically irrelevant (section 4).

10. This is fine. This chapter is about the SE for percentages, but, the idea of computing SEs for numbers should not disappear.

Chapter 21

1. (a) The sample is like 500 draws from a box with 25,000 tickets; each ticket is marked 1 (has a computer) or 0 (does not have a computer). The number of sample households with computers is like the sum of the draws.

The fraction of 1's in the box is unknown, but can be estimated by the fraction in the sample, as 79/500 = 0.158. On this basis, the SD of the box is estimated as » 0.36. The SE for the number of sample households with computers is estimated as x 0.36 » 8, and 8 out of 500 is 1.6%. The percentage of households in the town with computers is estimated as 15.8%, and the estimate is likely to be off by 1.6% or so.

(b) The 95%-confidence interval is 15.8% ± 3.2%.

Comment. You need to estimate the percentage for the town from data for the town: the national figures may not apply. (In this case, the town seems pretty close to the national average.)

4. (a) The box has millions of tickets, one for each 17-year-old in school that year. Tickets are marked 1 for those who knew that Chaucer wrote The Canterbury Tales, and 0 for the others. The data are like 6000 draws from the box, and the number of students in the sample who know the answer is like the sum of the draws. The fraction of 1 's in the box can be estimated from the sample as 0.361 On this basis, the SD of the box is estimated as » 0.48. The SE for the number of students in the sample who know the answer is estimated as x 0.48 » 37. The SE for the percentage is 37/6000, which is about 0.6 of 1%. The percentage of students in the population who know the answer is estimated as 36.1%, give or take 0.6 of 1 % or so. The 95%-confidence interval is 36.1% ± 1.2%.

(b) 95.2% ± 0.6 of 1%.

6. This is not the right SE. We do not have a simple random sample of 252 days, and the daily changes are dependent: each day's closing price is the next day's opening price.

Chapter 23

4. Can't be done with the information given. This is a simple random sample of households, but a cluster sample of people. The cluster is the household, and people in a household are likely to be similar with respect to commuting. For example, if a household is far from the center of town, all the occupants are likely to have a long commute. The SE is going to be bigger than the SE for a simple random sample of 2500 persons: section 22.5.

10. (a) True: the SE is estimated from the sample data, as on p.416.

(b) False. There is no such thing as a 95%-confidence interval for the sample

average, you know the sample average. It's the population average that you have to worry about (pp.385-86).

(c) True.

(d) False. This confuses the SD with the SE. And it's ridiculous, because a household must have a whole number of persons (1, or 2, or 3, and so forth).

(e) False. For instance, if household size followed the normal curve, there would be many households with a negative number of occupants; we're not ready for that.

(f) True. See pp. 411 and 418-19. Even though household size does not follow the normal curve, you can still use the normal curve to approximate the probability histogram for the sample average.

Chapter 23 Special Review Exercises

18. If a student answers at random, the score is like the sum of 50 draws from the box | 2 -1 -1 | . The average of this box is 0, and the SD is 1.41 by the shortcut (section 17.4). The EV for the sum of the draws is 50 x 0 = 0; the SE is x 1.41 » 10. The score will be around 0, give or take 10 or so.

(a) A cutoff of 50 is 5 SEs, the chance is zilch.

(b) A cutoff of 10 is 1 SE, the chance is about 16%.

DISC (Lab Manual)

9. The mean for the box is 0.2; the SD is 2.4. So you would expect a score of 0.2 x 100 questions = 20 just by chance alone plus or minus the squareroot of 100 x 2.4 = 10 x 2.4 = 24.