Miles on an Impala A random sample of 18 three-year-old Chevrolet Impalas was obtained

Chapter 15, Problem 12

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Miles on an Impala A random sample of 18 three-year-old Chevrolet Impalas was obtained in the St. Louis, Missouri area, and the number of miles on each car was recorded as follows: 27,647 30,858 67,476 35,874 39,943 55,702 31,739 31,832 38,194 29,949 36,231 67,097 32,707 25,672 45,109 26,199 60,798 66,592 Source: carsoup.com Test the hypothesis that the median number of miles on 3-yearold Impalas is less than 40,428 miles, the median miles driven over three years nationwide, at the a = 0.05 level of significance. Note: A normal probability plot indicates that the data are not normal. 1316 discuss the by-hand P-value approach to testing hypotheses regarding the median by using the sign test. We can use the P-value approach when determining whether or not to reject the null hypothesis regarding a median by using the sign test. Recall that the P-value is the probability of observing a test statistic as extreme or more extreme than what was actually observed, under the assumption that the null hypothesis is true. In the sign test, we assume that the median is M0, so 50% of the data should be less than M0 and 50% of the data greater than M0. So we expect half of the data to result in minus signs and half of the data to result in plus signs. We can think of the data as a bunch of plus and minus signs that follow a binomial probability distribution with p = 1 2 if the null hypothesis is true. So the P-value is computed from the binomial probability formula, with X = k and n equal to the number of plus and minus signs: P@value = P(X k) = nCk0.5k (1 - 0.5)n-k + nCk-10.5k-1 (1 - 0.5)n-(k-1) + g + nC0(1 - 0.5)n For Example 1 in this section, the P-value is P@value = P(X 8) = 20C8 # 0.58 # (1 - 0.5)20-8 + 20C7 # 0.57 # (1 - 0.5)13 + g + 20C0(1 - 0.5)20 = 0.2517 Because the P-value is greater than the level of significance, a = 0.05, we do not reject the null hypothesis. These binomial probabilities are easiest to compute with statistical software or a graphing calculator with advanced statistical features.