648 CHAPTER 13 Nonparametric Tests KEY ELEMENTS Sign Test Objective Use positive and negative signs to test a claim falling into one of the following three categories: 1. Matched Pairs • Subtract the second value in each pair from the first, record the sign of the difference, and ignore any 0s. 2. Nominal Data with Two Categories • Represent each member of one category by a positive sign and represent each member of the other category by a negative sign. 3. Median of a Single Population • Subtract the median from each sample value, record the sign of the difference, and ignore any 0s. Notation x = the number of times the less frequent sign occurs n = the total number of positive and negative signs combined Requirements The sample data are a simple random sample. Note: There is no requirement that the sample data come from a population with a particular distribution, such as a normal distribution. Test Statistic If n … 25: Test statistic is x = the number of times the less frequent sign occurs. If n 7 25: Test statistic is z = 1x + 0.52 - a n 2b 2 n 2 P-Values P-values are often provided by technology, or P-values can often be found using the z test statistic. Critical Values 1. If n … 25, critical x values are found in Table A-7. 2. If n 7 25, critical z values are found in Table A-2. Hint: Because x or z is based on the less frequent sign, all one-sided tests are treated as if they were left-tailed tests. CAUTION When using the sign test in a one-tailed test, be very careful to avoid making the wrong conclusion when one sign occurs significantly more often or less often than the other sign, but the sample data contradict the alternative hypothesis. A sample of 7% boys can never be used to support the claim that boys occur more than 50% of the time, as in Example 1.
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