690 CHAPTER 13 Nonparametric Tests Large Sample: Testing Temperatures for Randomness Above and Below the Mean EXAMPLE 2 Use the following mean global temperatures 1°C2 for 50 recent and consecutive years to test for randomness above and below the mean. Use a 0.05 significance level. The data are listed in order by row. 13.98 14.10 14.05 14.03 13.65 13.75 13.93 13.98 13.91 14.00 14.04 13.90 13.95 14.18 13.94 13.98 13.79 14.16 14.07 14.13 14.27 14.40 14.10 14.34 14.16 14.13 14.19 14.35 14.42 14.28 14.49 14.44 14.16 14.18 14.31 14.47 14.36 14.40 14.71 14.44 14.41 14.56 14.70 14.64 14.60 14.77 14.64 14.66 14.68 14.70 SOLUTION REQUIREMENT CHECK (1) The data are arranged in order. (2) Each data value can be categorized into one of two separate categories: below the mean or above the mean. The requirements are satisfied. The null and alternative hypotheses are as follows: H0: The sequence is random. H1: The sequence is not random. The mean of those 50 temperatures is 14.250°C. If we replace each temperature with B if it is below the mean and A if it is above the mean, we get the following sequence. For example, the first temperature of 13.98°C is below the mean of 14.250°C, so 13.98 is replaced with B. BBBBBBBBBB BBBBBBBBBB AABABBBAAA AABBAAAAAA AAAAAAAAAA Examination of the sequence of B’s and A’s shows that the letter B occurs 26 times, the letter A occurs 24 times, and the number of runs is 8, so we have the following: n1 = number of B>s = 26 n2 = number of A>s = 24 G = number of runs = 8 Since, n1 7 20, we need to calculate the test statistic z, so we must first evaluate mG and sG as follows: mG = 2n1n2 n1 + n2 + 1 = 212621242 26 + 24 + 1 = 25.96 sG = B12n1n2212n1n2 - n1 - n22 1n1 + n22 21n 1 + n2 - 12 = B122126212423212621242 - 26 - 244 126 + 2422126 + 24 - 12 = 3.49355558 We now find the test statistic: z = G - mG sG = 8 - 25.96 3.49355558 = -5.14 We can use the test statistic z = -5.14 to find that the P-value is 0.000, which is less than the significance level of a = 0.05, so we reject the null hypothesis of randomness.
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