8-2 Testing a Claim About a Proportion 397 Using the Exact Method EXAMPLE 2 In Example 1 we have n = 19,136 subjects and 29.2% of them (or 5588) have sleepwalked. In Example 1 we used a hypothesis test to determine whether 29.2% is significantly less than 30% (or to determine whether 5588 is significantly low). Here, we can use the binomial probability distribution to find the probability of getting 5588 or fewer sleepwalkers when 19,136 subjects are randomly selected. Shown here is a portion of the Statdisk results obtained when finding binomial probabilities with n = 19,136 and p = 0.30. This Statdisk result shows that the probability of 5588 sleepwalkers or fewer is 0.0080286, which is the P-value. (This is the P-value because it is the probability of getting a result “at least as extreme” as the result of 5588 sleepwalkers that was obtained.) Given that the P-value is 0.0080286 and the significance level is 0.05, we reject the null hypothesis as we did in Example 1. We support the alternative hypothesis and we conclude that there is sufficient evidence to support the claim that fewer than 30% of adults have sleepwalked. YOUR TURN. Do Exercise 33 “Exact Method.” Statdisk PART 2 Exact Methods for Testing Claims About a Population Proportion p Instead of using the normal distribution as an approximation to the binomial distribution, we can get exact results by using the binomial probability distribution itself. Binomial probabilities are a real nuisance to calculate manually, but technology makes this approach quite simple. Also, this exact approach does not require that np Ú 5 and nq Ú 5, so we have a method that applies when that requirement is not satisfied. To test hypotheses using the exact method, find P-values as follows: Exact Method Identify the sample size n, the number of successes x, and the claimed value of the population proportion p (used in the null hypothesis); then find the P-value by using technology for finding binomial probabilities as follows: Left-tailed test: P@value = P1x or fewer successes among n trials2 Right-tailed test: P@value = P1x or more successes among n trials2 Two-tailed test: P-value = twice the smaller of the preceding left-tailed and right-tailed values Note: There is no universally accepted method for the two-tailed exact case, so this case can be treated with other different approaches, some of which are quite complex. For example, Minitab uses a “likelihood ratio test” that is different from the above approach that is commonly used.
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