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CHAPTER 12 10
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Nonparametric Tests
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Introduction
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Most tests of hypotheses and significance (or decision rules) considered in previous chapters require various assumptions about the distribution of the population from which the samples are drawn. For example, in 5 the population distributions often are required to be normal or nearly normal. Situations arise in practice in which such assumptions may not be justified or in which there is doubt that they apply, as in the case where a population may be highly skewed. Because of this, statisticians have devised various tests and methods that are independent of population distributions and associated parameters. These are called nonparametric tests. Nonparametric tests can be used as shortcut replacements for more complicated tests. They are especially valuable in dealing with nonnumerical data, such as arise when consumers rank cereals or other products in order of preference.
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The Sign Test
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Consider Table 10-1, which shows the numbers of defective bolts produced by two different types of machines (I and II) on 12 consecutive days and which assumes that the machines have the same total output per day. We wish to test the hypothesis H0 that there is no difference between the machines: that the observed differences between the machines in terms of the numbers of defective bolts they produce are merely the result of chance, which is to say that the samples come from the same population. A simple nonparametric test in the case of such paired samples is provided by the sign test. This test consists of taking the difference between the numbers of defective bolts for each day and writing only the sign of the difference; for instance, for day 1 we have 47 71, which is negative. In this way we obtain from Table 10-1 the sequence of signs (1) (i.e., 3 pluses and 9 minuses). Now if it is just as likely to get a as a , we would expect to get 6 of each. The test of H0 is thus equivalent to that of whether a coin is fair if 12 tosses result in 3 heads ( ) and 9 tails ( ). This involves the binomial distribution of 4. Problem 10.1 shows that by using a two-tailed test of this distribution at the 0.05 significance level, we cannot reject H0; that is, there is no difference between the machines at this level. Remark 1 If on some day the machines produced the same number of defective bolts, a difference of zero would appear in sequence (1). In that case we can omit these sample values and use 11 instead of 12 observations. Remark 2 A normal approximation to the binomial distribution, using a correction for continuity, can also be used (see Problem 10.2).
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CHAPTER 10 Nonparametric Tests
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Table 10-1 Day Machine I Machine II 1 47 71 2 56 63 3 54 45 4 49 64 5 36 50 6 48 55 7 51 42 8 38 46 9 61 53 10 49 57 11 56 75 12 52 60
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Although the sign test is particularly useful for paired samples, as in Table 10-1, it can also be used for problems involving single samples (see Problems 10.3 and 10.4).
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The Mann Whitney U Test
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Consider Table 10-2, which shows the strengths of cables made from two different alloys, I and II. In this table we have two samples: 8 cables of alloy I and 10 cables of alloy II. We would like to decide whether or not there is a difference between the samples or, equivalently, whether or not they come from the same population. Although this problem can be worked by using the t test of 7, a nonparametric test called the Mann Whitney U test, or briefly the U test, is useful. This test consists of the following steps:
Table 10-2 Alloy I 18.3 18.9 16.4 25.3 22.7 16.1 17.8 24.2 12.6 19.6 14.1 12.9 Alloy II 20.5 15.2 10.7 11.8 15.9 14.7
Step 1. Combine all sample values in an array from the smallest to the largest, and assign ranks (in this case from 1 to 18) to all these values. If two or more sample values are identical (i.e., there are tie scores, or briefly ties), the sample values are each assigned a rank equal to the mean of the ranks that would otherwise be assigned. If the entry 18.9 in Table 10-2 were 18.3, two identical values 18.3 would occupy ranks 12 and 13 in the array so that 1 the rank assigned to each would be 2(12 13) 12.5. Step 2. Find the sum of the ranks for each of the samples. Denote these sums by R1 and R2, where N1 and N2 are the respective sample sizes. For convenience, choose N1 as the smaller size if they are unequal, so that N1 N2. A significant difference between the rank sums R1 and R2 implies a significant difference between the samples. Step 3. To test the difference between the rank sums, use the statistic U N1N2 N1(N1 2 1) R1 (2)
corresponding to sample 1. The sampling distribution of U is symmetrical and has a mean and variance given, respectively, by the formulas mU N1N2 2 s2 U N1N2(N1 12 N2 1) (3)
If N1 and N2 are both at least equal to 8, it turns out that the distribution of U is nearly normal, so that Z U mU sU (4)
is normally distributed with mean 0 and variance 1. Using Appendix C, we can then decide whether the samples are significantly different. Problem 10.5 shows that there is a significant difference between the cables at the 0.05 level.
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