Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Chapter 10Statistical Inference for Two SamplesApplied Statistics and Probability for EngineersSixth EditionDouglas C. Montgomery George C. RungerCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.210Statistical Inference for Two Samples10-1 Inference on the Difference in Means of Two Normal Distributions, Variances Known10-1.1 Hypothesis tests on the difference in means, variances known10-1.2 Type II error and choice of sample size10-1.3 Confidence interval on the difference in means, variance known10-2 Inference on the Difference in Means of Two Normal Distributions, Variance Unknown10-2.1 Hypothesis tests on the difference in means, variances unknown10-2.2 Type II error and choice of sample size10-2.3 Confidence interval on the difference in means, variance unknown10-3 A Nonparametric Test on the Difference in Two Means10-4 Paired t-Tests10-5 Inference on the Variances of Two Normal Populations10-5.1 F distributions10-5.2 Hypothesis tests on the ratio of two variances10-5.3 Type II error and choice of sample size10-5.4 Confidence interval on the ratio of two variances10-6 Inference on Two Population Proportions10-6.1 Large sample tests on the difference in population proportions10-6.2 Type II error and choice of sample size10-6.3 Confidence interval on the difference in population proportions10-7 Summary Table and Roadmap for Inference Procedures for Two SamplesCHAPTER OUTLINEChapter 10 Title and OutlineCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Learning Objectives for Chapter 10After careful study of this chapter, you should be able to do the following:1. Structure comparative experiments involving two samples as hypothesis tests.2. Test hypotheses and construct confidence intervals on the difference in means of two normal distributions.3. Test hypotheses and construct confidence intervals on the ratio of the variances or standard deviations of two normal distributions.4. Test hypotheses and construct confidence intervals on the difference in two population proportions.5. Use the P-value approach for making decisions in hypothesis tests.6. Compute power, Type II error probability, and make sample size decisions for two-sample tests on means, variances & proportions.7. Explain & use the relationship between confidence intervals and hypothesis tests.3Chapter 10 Learning ObjectivesCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.10-1: Inference on the Difference in Means of Two Normal Distributions, Variances KnownAssumptions41. Let be a random sample from population 1.2. Let be a random sample from population 2.3. The two populations X1and X2are independent.4. Both X1and X2 are normal.111211,,,nXXX 222221,,,nXXX Sec 10-1 Inference on the Difference in Means of Two Normal Distributions, Variances KnownCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.5The quantity(10-1)has a N(0, 1) distribution.2221212121)(nnXXZ10-1: Inference on the Difference in Means of Two Normal Distributions, Variances KnownSec 10-1 Inference on the Difference in Means of Two Normal Distributions, Variances KnownCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.10-1.1 Hypothesis Tests on the Difference in Means, Variances Known6Null hypothesis: H0: 1 2= 0Test statistic: (10-2)2221210210nnXXZAlternative Hypotheses P-Value Rejection Criterion For Fixed-Level TestsH0: 1 2≠ 0Probability above |z0| and probability below |z0|, P = 2[1 (|z0|)]z0 z2or z0 z2H1: 1 2> 0Probability above z0, P = 1 (z0)z0 zH1: 1 2< 0Probability below z0, P = (z0)z0 zSec 10-1 Inference on the Difference in Means of Two Normal Distributions, Variances KnownCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.EXAMPLE 10-1 Paint Drying Time7A product developer is interested in reducing the drying time of a primer paint. Two formulations of the paint are tested; formulation 1 is the standard chemistry, and formulation 2 has a new drying ingredient that should reduce the drying time. From experience, it is known that the standard deviation of drying time is 8 minutes, and this inherent variability should be unaffected by the addition of the new ingredient. Ten specimens are painted with formulation 1, and another 10 specimens are painted with formulation 2; the 20 specimens are painted in random order. The two sample average drying times are minutes and minutes, respectively. What conclusions can the product developer draw about the effectiveness of the new ingredient, using 0.05?The seven-step procedure is:1. Parameter of interest: The difference in mean drying times, 1 2, and 0 0.2. Null hypothesis: H0: 1 2 0.3. Alternative hypothesis: H1: 1 2.1211x1122xSec 10-1 Inference on the Difference in Means of Two Normal Distributions, Variances KnownCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.EXAMPLE 10-1 Paint Drying Time - Continued84. Test statistic: The test statistic iswhere and n1 n2 10.5. Reject H0if: Reject H0: 1 2if the P-value is less than 0.05.6. Computations: Since minutes and minutes, the test statistic is7. Conclusion: Since z0= 2.52, the P-value is P 1 (2.52) 0.0059, so we reject H0at the 0.05 levelInterpretation: We can conclude that adding the new ingredient to the paint significantly reduces the drying time.2221212100nnxxz64)8(222211211x2112x 52.2108108112121220zSec 10-1 Inference on the Difference in Means of Two Normal Distributions, Variances KnownCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.10-1.2 Type II Error and Choice of Sample SizeUse of Operating Characteristic CurvesTwo-sided alternative:One-sided alternative:9222102221021d222102221021dSec 10-1 Inference on the Difference in Means of Two Normal Distributions, Variances KnownCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.10-1.2 Type II Error and Choice of Sample SizeSample Size FormulasTwo-sided
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