STATISTIKA INDUSTRI 2 TIN 4004

Transcription

1 STATISTIKA INDUSTRI 2 TIN 4004

2 Pertemuan 11 & 12 Outline: Nonparametric Statistics Referensi: Walpole, R.E., Myers, R.H., Myers, S.L., Ye, K., Probability & Statistics for Engineers & Scientists, 9 th Ed. Prentice Hall, 2012.

3 Nonparametric Distribution-free methods Analysis of ranks Small sample size Disadvantages: Do not utilize all information provided by the sample Less efficient than parametric procedure

4 Sign Test Used to test hypotheses on a population median Population mean = population median when distribution is symmetric In testing the H 0 : μ = μ 0 against an appropriate alternative, with random sample size = n, replace each sample value exceeding μ 0 with +, and each sample value exceeding μ 0 with - The sign test is applicable only in situations where μ 0 cannot equal the value of any of the observations Binomial random variable X, representing the number of plus signs in our random sample

5 Sign Test Test H 0 that the number of + is a value of a random variable having the binomial distribution with p = 1/2. P-values are calculated using binomial distribution Reject H 0 if proportion of + is sufficiently less than ½, when the value x of our random variable is small. P value α

6 Sign Test Reject H 0, jika P-value α

7 Contoh: Sign Test

8 Contoh: Sign Test

9 Contoh Sign Test

10 Contoh Sign Test

11 Wilcoxon Signed-Rank Test Symmetric continuous distribution Subtract sample value with μ 0, rank it from absolute smallest to the largest one When there are more than one differences are the same, rank it with the average number of the differences

12 Wilcoxon Signed-Rank Test Test Procedures

13 Wilcoxon Signed-Rank Test n < 5, and level of significance 0,05 (onetailed test), level of significance 01 (twotailed test) >>> w +, w, w will lead to acceptance H 0 5 n 30, check table to set critical region

14 Contoh: Wilcoxon Signed-Rank Test

15 Contoh: Wilcoxon Signed-Rank Test

16 Contoh: Wilcoxon Signed-Rank Test

17 Wilcoxon Signed-Rank Test

18 Wilcoxon Rank-Sum Test Testing equality of means of two continous distributions that nonnormal and samples are independent Take random sample, assign n 1 for smaller number sample and n 2 for larger one. Assigned randomly if two population have the same number of sample Arrange n 1 + n 2 observations in ascending order. If there are the identical observations value, mean the ranks w 1 = sum of ranks of n 1 observations w 2 = sum of ranks of n 2 observations

19 Wilcoxon Rank-Sum Test

20 Wilcoxon Rank-Sum Test Procedures: Reject H 0 : u 1, u 2, u less than or equal to the table value

21 Contoh: Wilcoxon Rank-Sum Test

22 Wilcoxon Rank-Sum Test

23 Wilcoxon Rank-Sum Test

24 Kruskal-Wallis Test Nonparametric alternative to analysis of variance ANOVA: testing equality of k 2 population means, must be normal distribution when using F-statistic Kruskal-Wallis Test is a nonparametric procedure for testing the equality of means in the one-factor analysis of variance without normal populations assumption Generalization of Runk-Sum test for case k > 2 samples

25 Kruskal-Wallis Test Procedure: Test H 0 : μ 1 = μ 2 = = μ k ; H 1 : Not all means are equal Condition: samples are independent Steps: 1. Arrange the k samples in ascending order, and assigne the smallest number observations as n 1 and so on. Compute n = n 1 + n n k 2. Rank all the observations inascending order. For identical observations, assign it with the mean of the ranks 3. Sum the rank of each sample, denote it by random variable R. R i is sum of ranks corresponding to the n i observation in the i-th sample

26 Kruskal-Wallis Test Procedure: Steps: 4. Compute the H-statistic: H = 12 n(n + 1) k R i 2 n i 3(n + 1) i=1 >>> approximated very well by chi-squared distribution with df = k 1 5. Critical Region: 2 H > χ α,v=k 1

27 Contoh soal: Kruskal-Wallis Test

28 Contoh soal: Kruskal-Wallis Test

29 Runs Test Randomness Test Run: subsequence of one or more identical symbols representing a common property of the data Runs test divides the data into two mutually exclusive categories, so a sequence will always be limited to two distinct symbols n 1 : the number of symbols category that the least occurs; n 2 : the number of symbols belong to other category n = n 1 + n 2 Based on the random variable V V: total number of runs that occur in the complete sequence of experiment

30 Runs Test Hipotesis: H 0 : the sequence is random H 1 : the sequence is not random Tabel Runs Test to determine the P-value: One tailed test: P = P(V v, when H 0 is true) Two tailed test: P = 2P(V v, when H 0 is true) When v is large (> n/2), use: P = P V v, when H 0 is true = 1 P(V v 1, when H 0 is true) Critical Region: P value α

31 Contoh Soal: Runs Test

32 Contoh Soal: Runs Test

33 Runs Test When n 1 and n 2 ( 10 for each) is large, the sampling distribution of V approaches the normal distribution with mean and variance as follow: Z Test:

34 Runs Test Lakukan uji apakah data berikut random atau tidak:

35 Kolmogorov-Smirnov Test Test for normality An alternative to the chi-squared test for distribution hypothesis test

36 Kolmogorov-Smirnov Test

37 Kolmogorov-Smirnov Test

38 Pertemuan 14 - Persiapan Materi Validitas dan Realibilitas

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