Sign Test for Paired Samples

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The sign test can also be applied to paired observations drawn from continuous populations. Let (X1j, X2j), j 1, 2, . . . , n be a collection of paired observations from two continuous populations, and let Dj X1j X2j j 1, 2, . . . , n

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15-2 SIGN TEST

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be the paired differences. We wish to test the hypothesis that the two populations have a common median, that is, that 1 2. This is equivalent to testing that the median of the differences D 0. This can be done by applying the sign test to the n observed differences dj, as illustrated in the following example.

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EXAMPLE 15-3

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An automotive engineer is investigating two different types of metering devices for an electronic fuel injection system to determine whether they differ in their fuel mileage performance. The system is installed on 12 different cars, and a test is run with each metering device on each car. The observed fuel mileage performance data, corresponding differences, and their signs are shown in Table 15-2. We will use the sign test to determine whether the median fuel mileage performance is the same for both devices using 0.05. The eightstep-procedure follows: 1. 2. 3. 4. 5. 6. 7. 8. The parameters of interest are the median fuel mileage performance for the two metering devices. H0: H1:

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1 1 2, 2,

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or, equivalently, H0: or, equivalently, H1:

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0.05 We will use Appendix Table VII to conduct the test, so the test statistic is r min(r , r ). Since 0.05 and n 12, Appendix Table VII gives the critical values as r*.05 2. 0 We will reject H0 in favor of H1 if r 2. Computations: Table 15-2 shows the differences and their signs, and we note that r 8, r 4, and so r min(8, 4) 4. Conclusions: Since r 4 is not less than or equal to the critical value r*.05 2, we 0 cannot reject the null hypothesis that the two devices provide the same median fuel mileage performance.

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Table 15-2 Performance of Flow Metering Devices Metering Device Car 1 2 3 4 5 6 7 8 9 10 11 12 1 17.6 19.4 19.5 17.1 15.3 15.9 16.3 18.4 17.3 19.1 17.8 18.2 2 16.8 20.0 18.2 16.4 16.0 15.4 16.5 18.0 16.4 20.1 16.7 17.9 Difference, dj 0.8 0.6 1.3 0.7 0.7 0.5 0.2 0.4 0.9 1.0 1.1 0.3 Sign

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578 15-2.3

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CHAPTER 15 NONPARAMETRIC STATISTICS

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Type II Error for the Sign Test

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The sign test will control the probability of type I error at an advertised level for testing the null hypothesis H0: for any continuous distribution. As with any hypothesis-testing procedure, it is important to investigate the probability of a type II error, . The test should be able to effectively detect departures from the null hypothesis, and a good measure of this effectiveness is the value of for departures that are important. A small value of implies an effective test procedure. In determining , it is important to realize not only that a particular value of , say 0 , must be used but also that the form of the underlying distribution will affect the calculations. To illustrate, suppose that the underlying distribution is normal with 1 and we are testing the hypothesis H0: in the normal distribution, this is equiv2 versus H1: 2. (Since alent to testing that the mean equals 2.) Suppose that it is important to detect a departure from 2 to 3. The situation is illustrated graphically in Fig. 15-1(a). When the alternative hypothesis is true (H1: 3), the probability that the random variable X is less than or equal to the value 2 is p P1X 22 P1Z 12 1 12 0.1587

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Suppose we have taken a random sample of size 12. At the 0.05 level, Appendix Table VII r* 2. Therefore, is the probability that indicates that we would reject H0: 2 if r 0.05 we do not reject H0: 2 when in fact 3, or 1 12 x 12 a a x b 10.15872 10.84132 x 0

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=1

=1

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Under H0 : = 2 (a)

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Under H1 : = 3

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Figure 15-1 Calculation of for the sign test. (a) Normal distributions. (b) Exponential distributions.

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=2

= 2.89

= 4.33

Under H0 : = 2 (b)

Under H1 : = 3