1. Pengujian Hipotesis Dua Populasi By. Nurvita Arumsari, Ssi, MSi

2. After completing this chapter, you should be able to:  Test hypotheses or form interval estimates for ◦ two independent population means  Standard deviations known  Standard deviations unknown ◦ two means from paired samples ◦ the difference between two population proportions

3. Estimating two population values Population means, independent samples Paired samplesPopulation proportions Group 1 vs. independent Group 2 Same group before vs. after treatment Proportion 1 vs. Proportion 2 Examples:

5. Lower tail test: H 0 : μ 1 = μ 2 H A : μ 1 < μ 2 i.e., H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 < 0 Upper tail test: H 0 : μ 1 = μ 2 H A : μ 1 > μ 2 i.e., H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 > 0 Two-tailed test: H 0 : μ 1 = μ 2 H A : μ 1 ≠ μ 2 i.e., H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 ≠ 0 Two Population Means, Independent Samples

6. Lower tail test: H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 < 0 Upper tail test: H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 > 0 Two-tailed test: H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 ≠ 0  /2  -z  - z  /2 zz z  /2 Reject H 0 if z < - z  Reject H 0 if z > z  Reject H 0 if z < - z  /2  or z > z  /2

7. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Use a z test statistic Use s to estimate unknown σ, approximate with a z test statistic Use s to estimate unknown σ, use a t test statistic and pooled standard deviation

8. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30  Different data sources ◦ Unrelated ◦ Independent  Sample selected from one population has no effect on the sample selected from the other population  Use the difference between 2 sample means  Use z test or pooled variance t test *

10. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Assumptions:  Samples are randomly and independently drawn  population distributions are normal or both sample sizes are  30  Population standard deviations are known *  When σ 1 and σ 2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a z-value…

11. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The test statistic for μ 1 – μ 2 is: *

12. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The confidence interval for μ 1 – μ 2 is: (continued) *

14. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Assumptions:  Samples are randomly and independently drawn  both sample sizes are  30  Population standard deviations are unknown *

15. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Forming interval estimates:  use sample standard deviation s to estimate σ  the test statistic is a z value (continued) *

16. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The confidence interval for μ 1 – μ 2 is: (continued) *

18. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Assumptions:  populations are normally distributed  the populations have equal variances  samples are independent *

19. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Forming interval estimates:  The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ  the test statistic is a t value with (n 1 + n 2 – 2) degrees of freedom (continued) *

20. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The pooled standard deviation is (continued) *

21. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The confidence interval for μ 1 – μ 2 is: (continued) Where t  /2 has (n 1 + n 2 – 2) d.f., and *

22. Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Where t  /2 has (n 1 + n 2 – 2) d.f., and The test statistic for μ 1 – μ 2 is: *

24. Lower tail test: H 0 : μ d = 0 H A : μ d < 0 Upper tail test: H 0 : μ d = 0 H A : μ d > 0 Two-tailed test: H 0 : μ d = 0 H A : μ d ≠ 0 Paired Samples  /2  -t  -t  /2 tt t  /2 Reject H 0 if t < - t  Reject H 0 if t > t  Reject H 0 if t < - t   or t > t  Where t has n - 1 d.f. (continued)

25. Tests Means of 2 Related Populations  Paired or matched samples  Repeated measures (before/after)  Use difference between paired values:  Eliminates Variation Among Subjects  Assumptions: ◦ Both Populations Are Normally Distributed ◦ Or, if Not Normal, use large samples d = x 1 - x 2

26. The i th paired difference is d i, where Paired samples d i = x 1i - x 2i The point estimate for the population mean paired difference is d : The sample standard deviation is n is the number of pairs in the paired sample

27. The confidence interval for d is Paired samples Where t  /2 has n - 1 d.f. and s d (continued) n is the number of pairs in the paired sample

29. Goal: Form a confidence interval for or test a hypothesis about the difference between two population proportions, p 1 – p 2 The point estimate for the difference is p 1 – p 2 Population proportions Assumptions: n 1 p 1  5, n 1 (1-p 1 )  5 n 2 p 2  5, n 2 (1-p 2 )  5

30. Population proportions Lower tail test: H 0 : p 1 – p 2 = 0 H A : p 1 – p 2 < 0 Upper tail test: H 0 : p 1 – p 2 = 0 H A : p 1 – p 2 > 0 Two-tailed test: H 0 : p 1 – p 2 = 0 H A : p 1 – p 2 ≠ 0  /2  -z  - z  /2 zz z  /2 Reject H 0 if z < -z  Reject H 0 if z > z  Reject H 0 if z < -z   or z > z 

31. Population proportions The pooled estimate for the overall proportion is: where x 1 and x 2 are the numbers from samples 1 and 2 with the characteristic of interest Since we begin by assuming the null hypothesis is true, we assume p 1 = p 2 and pool the two p estimates

32. Population proportions The test statistic for p 1 – p 2 is: (continued)

33. Population proportions The confidence interval for p 1 – p 2 is:

35. You’re a welding inspector. Is there a difference in level of precision between welding machine A & B? You collect the following data: A B Number 21 25 Sample mean 3.27 2.53 Sample std dev 1.30 1.16 Assuming equal variances, is there a difference in average yield (  = 0.05)?

36. The test statistic is:

37. H 0 : μ 1 - μ 2 = 0 i.e. (μ 1 = μ 2 ) H A : μ 1 - μ 2 ≠ 0 i.e. (μ 1 ≠ μ 2 )  = 0.05 df = 21 + 25 - 2 = 44 Critical Values: t = ± 2.0154 Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence of a difference in means. t 0 2.0154-2.0154.025 Reject H 0.025 2.040

38.  Assume you are a welding inspector. In the first inspection, you found some defects in welding result. Then, you got a welding training. After you got a training, you inspect the welding result, like in table. Is the training effective? You collect the following data: Paired Samples Example Welding ins.Number of DefectsDifference BeforeAfter C.B203-17 T.F206-14 M.H154-11 R.K104-6 M.O177-10

39.  Assume you are a welding inspector. In the first inspection, you found some defects in welding result. Then, you got a welding training. After you got a training, you inspect the welding result, like in table. Is the training effective? You collect the following data: d =  didi n = -11.2 Welding ins.Number of DefectsDifference BeforeAfter C.B203-17 T.F206-14 M.H154-11 R.K104-6 M.O177-10 Paired Samples Example

40. Is there a significant difference between the proportion of boy and the proportion of girl who will prefer on Welding department?  In a random sample, 52 of 72 boys and 10 of 50 girls indicated they would prefer welding department  Test at the.05 level of significance

41.  The hypothesis test is: H 0 : p 1 – p 2 = 0 (the two proportions are equal) H A : p 1 – p 2 ≠ 0 (there is a significant difference between proportions) The sample proportions are: Boys: p 1 = 52/72 =.72 Girls: p 2 = 10/50 =.2  The pooled estimate for the overall proportion is: Example: Two population Proportions (continued)