284 CHAPTER 6 Normal Probability Distributions CENTRAL LIMIT THEOREM For all samples of the same size n with n 7 30, the sampling distribution of x can be approximated by a normal distribution with mean m and standard deviation s>2n. Boston Commute Times EXAMPLE 1 Figures 6-21 and 6-22 illustrate the central limit theorem. ■ Original Data: Figure 6-21 is a histogram of 1000 Boston commute times (minutes) from Data Set 31 “Commute Times” in Appendix B. ■ Sample Means: Figure 6-22 is a histogram of 1000 sample means, where each of the 1000 samples includes 50 Boston commute times (randomly selected from Data Set 31 “Commute Times” in Appendix B). INTERPRETATION The original Boston commute times depicted in Figure 6-21 have a skewed distribution, but when we collect samples and compute their means, those sample means tend to have a distribution that is normal, as in Figure 6-22. FIGURE 6-21 Nonnormal Distribution: 1000 Boston Commute Times FIGURE 6-22 Approximately Normal Distribution of 1000 Sample Means A Universal Truth Example 1 and the central limit theorem are truly remarkable because they describe a rule of nature that works throughout the universe. If we could send a spaceship to a distant planet “in a galaxy far, far away,” and if we collect samples of rocks (all of the same large sample size) and weigh them, the sample means would have a distribution that is approximately normal. Think about the significance of that! The following key points form the foundation for estimating population parameters and hypothesis testing—topics discussed at length in the following chapters.
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