686 CHAPTER 13 Nonparametric Tests 17.Finding Critical Values An alternative to using Table A-9 to find critical values for rank correlation is to compute them using this approximation: rs = {B t2 t2 + n - 2 Here, t is the critical t value from Table A-3 corresponding to the desired significance level and n -2 degrees of freedom. Use this approximation to find critical values of rs for Exercise 15 “Ages of Best Actresses and Best Actors.” How do the resulting critical values compare to the critical values that would be found by using Formula 13-1 on page 678? 13-6 Beyond the Basics Key Concept This section describes the runs test for randomness, which is used to determine whether a sequence of sample data has a random order. This test requires a criterion for categorizing each data value into one of two separate categories, and it analyzes runs of those two categories to determine whether the runs appear to result from a random process, or whether the runs suggest that the order of the data is not random. 13-7 Runs Test for Randomness DEFINITIONS After characterizing each data value as one of two separate categories, a run is a sequence of data having the same characteristic; the sequence is preceded and followed by data with a different characteristic or by no data at all. The runs test uses the number of runs in a sequence of sample data to test for randomness in the order of the data. Fundamental Principle of the Runs Test Here is the key idea underlying the runs test: Reject randomness if the number of runs is very low or very high. ■ Example: The sequence of genders FFFFFMMMMM is not random because it has only 2 runs, so the number of runs is very low. ■ Example: The sequence of genders FMFMFMFMFM is not random because there are 10 runs, which is very high. The exact criteria for determining whether a number of runs is very high or low are found in the Key Elements box. The procedure for the runs test for randomness is also summarized in Figure 13-5. CAUTION The runs test for randomness is based on the order in which the data occur; it is not based on the frequency of the data. For example, a sequence of 3 men and 20 women might appear to be random, but the issue of whether 3 men and 20 women constitute a biased sample (with disproportionately more women) is not addressed by the runs test.
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