13-2 Sign Test 653 In Example 4, the sign test of the claim that the median is below 98.6°F results in a test statistic of z = -4.61 and a P-value of 0.00000202. However, a parametric test of the claim that m 6 98.6°F results in a test statistic of t = -6.611 with a P-value of 0.000000000813. Because the P-value from the sign test is not as low as the P-value from the parametric test, we see that the sign test isn’t as sensitive as the parametric test. Both tests lead to rejection of the null hypothesis, but the sign test doesn’t consider the sample data to be as extreme, partly because the sign test uses only information about the direction of the data, ignoring the magnitudes of the data values. The next section introduces the Wilcoxon signed-ranks test, which largely overcomes that disadvantage. Rationale for the Test Statistic Used When n + 25 When finding critical values for the sign test, we use Table A-7 only for n up to 25. When n 7 25, the test statistic z is based on a normal approximation to the binomial probability distribution with p = q = 1>2. Section 6-6 (available at www.TriolaStats.com) shows that the normal approximation to the binomial distribution is acceptable when both np Ú 5 and nq Ú 5. In Section 5-2 we saw that m = np and s = 1npq for binomial probability distributions. Because this sign test assumes that p = q = 1>2, we meet the np Ú 5 and nq Ú 5 prerequisites whenever n Ú 10. Also, with the assumption that p = q = 1>2, we get m = np = n>2 and s = 1npq = 1n>4 = 1n>2, so the standard z score z = x - m s becomes z = x - a n 2b 2 n 2 z 5 0 Reject Median 5 98.68 Fail to reject Median 5 98.68 Sample data: z 5 24.61 z 5 21.645 FIGURE 13-3 Testing the Claim That the Median Is Less Than 98.6°F INTERPRETATION There is sufficient sample evidence to support the claim that the median body temperature of healthy adults is less than 98.6°F. It is not equal to 98.6°F, as is commonly believed. YOUR TURN. Do Exercise 13 “Body Temperatures.” continued
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