Statistics Demystified, 2nd edition

Stan Gibilisco

Explanations for Quiz Answers in Chapter 5

1. After we sample an element of a set, we can either throw the sampled element away or else return it to the set. If we return the element to the set, we perform sampling with replacement. With this method, the set stays the same size after each sampling, no matter what the initial size of the set, and no matter how many times we take a sample. The correct choice is D.

2. If we narrow down the confidence interval in a normal distribution, we reduce the probability that any element selected "at random" will lie within that interval. It's as if we're taking smaller and smaller slices of a blueberry pie. The narrower the slice, the fewer berries we'll get. The correct choice is A.

3. The usefulness or validity of a sampling frame varies in proportion to its size relative to the whole population. The smaller the sampling frame, the less well (or the more poorly!) it will represent the population. The answer is D.

4. In a normal distribution, we define confidence intervals in terms of the number of standard deviations that we're allowed to "wander" below the mean and above the mean. If we can go up to one standard deviation above and below the mean, for example, we get a 68% confidence interval (as shown in Fig. 5-8 on page 177). If we can go up to two standard deviations above and below the mean, we get a 95% confidence interval (as shown in Fig. 5-9 on page 179). In order to define a confidence interval of any size, we must always know the mean and the standard deviation. The correct choice is C.

5. If we, as statisticians, evaluate data gathered by somebody else (and we don't do any of the experiments ourselves), we work with so-called secondary data. Neither choice A, choice B, nor choice C in this question say anything like that, so we must answer with D, "None of the above."

6. By definition, the 95% confidence interval in a normal distribution spans values within plus or minus two standard deviations of the mean. The correct choice is C.

7. If we want to do a good job of determining a confidence interval in a normal distribution, especially if the confidence interval is large, we must keep the span of values small compared with the estimate of the mean. You can review this principle by rereading the "Don't Be Deceived!" section on pages 182 and 183. The correct choice is A.

8. When we talk about the most bias, we refer to the sample that we would find the least objective as a source of data for the experiment at hand. People on the swimming team (choice A) would all get a great deal of exercise; they couldn't avoid it if they wanted to be good swimmers. That sample, therefore, would contain a lot of inherent bias. The street of residence (choice B), the month of birth (choice C), and the length of time that a person has lived in Hoodooburg (choice D) would have relatively little to do with the amount of exercise they get, certainly a lot less than whether or not they were on the swimming team! The answer to this question is A.

9. We're dealing with a huge population: 1,000,000,000,000 elements, which is hundreds of times the number of people in the world. We should expect that the smaller the sample, the less meaningful (or the "worse") the results of any particular experiment would be. If we test only 100 elements, no matter which 100, we're looking at only a hundred-millionth of one percent of the population! That's the smallest sample mentioned here, so we should expect that it would offer the least meaningful (or the "worst") results. The answer is D.

10. To answer this question, we can reread the explanation of the Central Limit Theorem given on page 176. For any normal distribution, the sampling distribution of means is also normal. The correct choice is B.