192 CHAPTER 4 Probability The simulation method used in Example 1 can be generalized as follows. Simulation Method for Testing a Claim About a Population Mean 1. Assume that the population mean is the value that is being claimed. Obtain the sample statistics of n, x, s, and identify the distribution (such as normal). Use technology to randomly generate many samples with these properties: The sample size is n (same as the sample being used), the standard deviation is s (the same standard deviation as the sample being used), and use the same distribution (such as normal) that was identified from the sample. For the mean, use the assumed (or claimed) value of the population mean. 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 value of x from the sample can easily occur or whether the value of x is significantly low or significantly high (because it rarely occurs in the randomly generated samples). Hint: Sort the means from the randomly generated samples. 4. If it appears that a sample mean of x is significantly low or significantly high, then its occurrence suggests that the assumed mean is likely to be incorrect. The following example illustrates the use of a simulation for finding a probability that would be very difficult to find using the methods presented in the preceding sections of this chapter. Examine this list of 50 simulated sample means with this question in mind: “How common is the Data Set 5 sample mean of 98.2°F?” The list of 50 sample means shows that if the population mean is actually 98.6°F as assumed, then such means will typically fall between 98.5°F and 98.7°F, so the sample mean of 98.2°F appears to be significantly low. It appears that if the population mean is actually 98.6°F, then a sample mean of 98.2°F is highly unlikely. Probability Value We can also use the 50 sample means listed above to estimate the probability of getting a sample mean of 98.20°F or lower, assuming that the population mean is equal to 98.6°F. Because none of the 50 sample means is 98.2°F or lower, the estimated probability of getting a sample mean of 98.2°F or lower is 0>50 = 0. Using more advanced methods, the actual probability can be found to be 0.00000000002. YOUR TURN. Do Exercise 9 “Body Temperatures.” INTERPRETATION With the assumption that the mean body temperature is 98.6°F, we have found that the sample mean of 98.2°F is highly unlikely and is significantly low. Because we did get the sample mean of 98.2°F from Data Set 5, we have strong evidence suggesting that the assumed population mean of 98.6°F is likely to be wrong! Probability of Three Birthdays That Are the Same EXAMPLE 2 Find the probability that among 100 randomly selected people, at least three share the same birthday. SOLUTION Although the probability problem is easy to state and understand, its solution can be quite difficult using the methods discussed in the previous sections of this chapter. Instead, a simulation can be used whereby different samples of 100 birthdays are
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