288 CHAPTER 6 Normal Probability Distributions Example 2 shows that we can use the same basic procedures from Section 6-2, but we must remember to correctly adjust the standard deviation when working with a sample mean instead of an individual sample value. The calculations used in Example 2 are exactly the type of calculations used by engineers when they design elevators, ski lifts, escalators, airplanes, boats, amusement park rides, and other devices that carry people. Introduction to Hypothesis Testing Carefully examine the conclusions that are reached in the next example illustrating the type of thinking that is the basis for the important procedure of hypothesis testing (formally introduced in Chapter 8). Example 3 uses the rare event rule for inferential statistics, first presented in Section 4-1: Identifying Significant Results with Probabilities: The Rare Event Rule for Inferential Statistics If, under a given assumption, the probability of a particular observed event is very small and the observed event occurs significantly less than or significantly greater than what we typically expect with that assumption, we conclude that the assumption is probably not correct. The following example illustrates the above rare event rule and it uses the author’s alltime favorite data set. This example also illustrates the type of reasoning that is used for the important method of hypothesis testing, which is formally introduced in Chapter 8. Body Temperatures EXAMPLE 3 Assume that the population of human body temperatures has a mean of 98.6°F, as is commonly believed. Also assume that the population standard deviation is 0.62°F (based on Data Set 5 “Body Temperatures” in Appendix B). A sample of size n = 106 subjects was randomly selected and the mean body temperature of 98.2°F was obtained. If the mean body temperature is really 98.6°F, find the probability of getting a sample mean of 98.2°F or lower for a sample of size 106. Based on the result, is 98.2°F significantly low? What do these results suggest about the common belief that the mean body temperature is 98.6°F? SOLUTION We work under the assumption that the population of human body temperatures has a mean of 98.6°F. We weren’t given the distribution of the population, but because the sample size n = 106 exceeds 30, we use the central limit theorem and conclude that the distribution of sample means is a normal distribution with these parameters: mx = m = 98.6 (by assumption) s x = s2 n = 0.62 2 106 = 0.0602197 Figure 6-24 shows the shaded area (see the tiny left tail of the graph) corresponding to the probability we seek. Having already found the parameters that apply to the distribution shown in Figure 6-24, we can now find the shaded area by using the same procedures developed in Section 6-2. Technology: If we use technology to find the shaded area in Figure 6-24, we get 0.0000000000155, which can be expressed as 0+.
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