4-5 Simulations for Hypothesis Tests 191 Here are two good uses of simulations: 1. Test a Claim Use a simulation to test some claim about a population parameter. (Example: Use the body temperatures from Data Set 5 “Body Temperatures” to test the common belief that the mean body temperature is 98.6°F. See Example 1.) 2. Find a Probability Use a simulation to find a probability that would be difficult to find using the methods presented earlier in this chapter. (Example: Find the probability that among 100 randomly selected people, at least three of them share the same birthday. See Example 2.) Test the Claim That the Mean Body Temperature is 98.6°F EXAMPLE 1 Data Set 5 “Body Temperatures” includes samples of body temperatures. Using the last column of body temperatures, we get the following: n = 106 x = 98.20°F s = 0.62°F Distribution: Approximately normal If humans have a mean body temperature of 98.6°F as is commonly believed, is the above sample result of x = 98.20°Fsignificantly low? If x = 98.20°F is significantly low, what does that suggest about the common belief that the mean body temperature is 98.6°F? Simulation Method Used in This Example: 1. Assume that the mean body temperature is 98.6°F and randomly generate many samples of 106 body temperatures from a normal distribution. (Random generators in technology typically require that we specify the normal distribution, the mean, the standard deviation, and the sample size. For the mean, enter the assumed value of 98.6, for the standard deviation enter the value of 0.62 from the sample in Data Set 5, and enter 106 for the sample size.) 2. Find the mean x for each generated sample and construct a list of those sample means. 3. Examine the list of sample means to see whether the sample mean of 98.2°F can easily occur or whether a value such as 98.2°F is very unlikely, so that 98.2°F appears to be significantly low. 4. If it appears that a sample mean such as 98.2°F is significantly low, then its actual occurrence suggests that the assumed mean of 98.6°F is likely to be incorrect. (Chapter 8 will introduce another procedure for testing the belief that 98.6°F is the mean body temperature.) SOLUTION Use a technology to repeat the process of randomly generating samples from a normally distributed population having the assumed mean of 98.6°F, the standard deviation of 0.62°F, and the sample size of n = 106. Obtain the mean of each generated sample. Using the above procedure, we obtain 50 sample means that have been sorted and listed below. 98.5 98.5 98.5 98.5 98.5 98.5 98.5 98.5 98.5 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.7 98.7 98.7 98.7 98.7 98.7 98.7 98.7 98.7 98.7 98.7 98.7 continued ) Cheating the Lottery Eddie Tipton was hired as director of security for the MultiState Lottery Association, even though he was a known convicted felon. Tipton managed to write and install a computer program for generating “random” lottery numbers, and this allowed him to reduce his odds against winning from 5 million to 1 down to 200 to 1. He was then able to acquire more than $24 million in winnings. He was caught and sentenced to up to 25 years in prison. Better luck next time, Eddie!
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