# ssrs 2012 barcode font Nonparametric Tests in Software Generating QR Code in Software Nonparametric Tests

CHAPTER 10 Nonparametric Tests
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(a) In this case there are 5 4 3 2 120 possibilities for choosing values for the two samples and the method of Problem 10.9 is too laborious. To simplify the procedure, let us concentrate on the smaller sample (of size N1 2) and the possible sums of the ranks, R1. The sum of the ranks for sample 1 is the smallest when the sample consists of the two lowest-ranking numbers (1, 2): then R1 1 2 3. Similarly, the sum of the ranks for sample 1 is the largest when the sample consists of the two highestranking numbers (4, 5); then R1 4 5 9. Thus R1 varies from 3 to 9. Column 1 of Table 10-17 lists these values of R1 (from 3 to 9), and column 2 shows the corresponding sample 1 values, whose sum is R1. Column 3 gives the frequency (or number) of samples with sum R1; for example, there are f 2 samples with R1 5. Since N1 2 and N2 3, we have U N1N2 N1(N1 2 1) R1 (2)(3) (2)(3) 2 R1 9 R1
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The probability that U R1 (i.e., Pr{U R1}) is shown in column 5 of Table 10-17 and is obtained by finding the relative frequency. The relative frequency is found by dividing each frequency f by the sum of 2 all the frequencies, or 10; for example, Pr5U 56 0.2. 10
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Table 10-17
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R1 3 4 5 6 7 8 9 Sample 1 Values (1, 2) (1, 3) (1, 4), (2, 3) (1, 5), (2, 4) (2, 5), (3, 4) (3, 5) (4, 5) f 1 1 2 2 2 1 1 U 6 5 4 3 2 1 0 Pr{U 0.1 0.1 0.2 0.2 0.2 0.1 0.1 R1}
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(b) From columns 3 and 4 of Table 10-17 we have mU s2 U # U a fU af a f (U af (1)(6 3)2 3 Another method s2 U U2 # U2 (1)(6)2 (1)(5)2 2 and N2 3 (2)(4)2 3, we have s2 U N1N2(N1 12 N2 1) (2)(3)(6) 12 3 (2)(3)2 10 (2)(2)2 (1)(1)2 (1)(0)2 (3)2 3 (1)(6) # U )2 (1)(5 3)2 (2)(4 3)2 (2)(3 3)2 10 (2)(2 3)2 (1)(1 3)2 (1)(0 3)2 (1)(5) 1 (2)(4) (2)(3) (2)(2) (1)(1) 1 2 2 2 1 1 (1)(0) 3
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(c) By formulas (3), using N1 mU N1N2 2
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(2)(3) 2
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10.12. If N numbers in a set are ranked from 1 to N, prove that the sum of the ranks is [N(N
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Let R be the sum of the ranks. Then we have R R 1 N 2 (N 3 1) c (N (N 2) 1) c N 2 1
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1)]>2.
(16) (17)
CHAPTER 10 Nonparametric Tests
where the sum in equation (17) is obtained by writing the sum in (16) backward. Adding equations (16) and (17) gives 2R (N 1) (N 1) (N 1) c (N 1) (N 1) N(N 1)
since (N 1) occurs N times in the sum; thus R [N(N 1)]>2. This can also be obtained by using a result from elementary algebra on arithmetic progressions and series.
10.13. If R1 and R2 are the respective sums of the ranks for samples 1 and 2 in the U test, prove that R1 R2 [N(N 1)]>2.
We assume that there are no ties in the sample data. Then R1 must be the sum of some of the ranks (numbers) in the set 1, 2, 3, . . . , N, while R2 must be the sum of the remaining ranks in the set. Thus the sum R1 R2 must be the sum of all the ranks in the set; that is, R1 R2 1 2 3 c N [N(N 1)]>2 by Problem 10.12.
The Kruskal Wallis H test 10.14. A company wishes to purchase one of five different machines: A, B, C, D, or E. In an experiment designed to determine whether there is a performance difference between the machines, five experienced operators each work on the machines for equal times. Table 10-18 shows the number of units produced by each machine. Test the hypothesis that there is no difference between the machines at the (a) 0.05, (b) 0.01 significance levels.
Table 10-18 A B C D E 68 72 60 48 64 72 53 82 61 65 77 63 64 57 70 42 53 75 64 68 53 48 72 50 53 A B C D E 17.5 21 10 2.5 14 21 6.5 25 11 16