280 CHAPTER 6 Normal Probability Distributions Why Sample with Replacement? All of the examples in this section involved sampling with replacement. Sampling without replacement would have the very practical advantage of avoiding wasteful duplication whenever the same item is selected more than once. Many of the statistical procedures discussed in the following chapters are based on the assumption that sampling is conducted with replacement because of the following two very important reasons. Reasons for Sampling with Replacement 1. When selecting a relatively small sample from a large population, it makes no significant difference whether we sample with replacement or without replacement. 2. Sampling with replacement results in independent events that are unaffected by previous outcomes, and independent events are easier to analyze and result in simpler calculations and formulas. TABLE 6-4 Sampling Distribution of Range Sample Sample Range Probability 4, 4 0 1>9 4, 5 1 1>9 4, 9 5 1>9 5, 4 1 1>9 5, 5 0 1>9 5, 9 4 1>9 9, 4 5 1>9 9, 5 4 1>9 9, 9 0 1>9 b. The last two columns of Table 6-4 list the values of the range along with the corresponding probabilities, so the last two columns constitute a table summarizing the probability distribution. Table 6-4 therefore describes the sampling distribution of the sample range. c. The mean of the sample ranges in Table 6-4 is 20>9 or 2.2. The population of 54, 5, 96 has a range of 9 - 4 = 5. Because the mean of the sample ranges (2.2) is not equal to the population range (5), the sample ranges do not target the value of the population range. d. Because the sample ranges do not target the population range, the sample range is a biased estimator of the population range. YOUR TURN. Do Exercise 13 “Sampling Distribution of the Range.” INTERPRETATION Because the sample range is a biased estimator of the population range, a sample range should generally not be used to estimate the value of the population range. Statistical Literacy and Critical Thinking 1.Fatal Car Crashes There are about 15,000 car crashes each day in the United States, and the proportion of car crashes that are fatal is 0.00559 (based on data from the National Highway Traffic Safety Administration). Assume that each day, 1000 car crashes are randomly selected and the proportion of fatal car crashes is recorded. a. What do you know about the mean of the sample proportions? b. What do you know about the shape of the distribution of the sample proportions? 6-3 Basic Skills and Concepts
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