1 10 Statistical Inference for Two Samples 10-1 Inference on the Difference in Means of Two Normal Distributions, Variances Known Hypothesis tests on the difference of means, variances known Type II error and choice of sample size Confidence interval on the difference in means, variance known 10-2 Inference on the Difference in Means of Two Normal Distributions, Variance Unknown Hypothesis tests on the difference of means, variances unknown Type II error and choice of sample size Confidence interval on the difference in means, variance unknown 10-3 A Nonparametric Test on the Difference of Two Means 10-4 Paired t-Tests 10-5 Inference on the Variances of Two Normal Populations F distributions Hypothesis tests on the ratio of two variances Type II error and choice of sample size Confidence interval on the ratio of two variances 10-6 Inference on Two Population Proportions Large sample tests on the difference in population proportions Type II error and choice of sample size Confidence interval on the difference in population proportions 10-7 Summary Table and Roadmap for Inference Procedures for Two Samples CHAPTER OUTLINE

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Learning Objectives for Chapter 10 After 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. 2

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-1: Introduction 3

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Figure 10-1 Two independent populations. 4

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Assumptions 5

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known 6

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Hypothesis Tests for a Difference in Means, Variances Known 7

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Type II Error and Choice of Sample Size Use of Operating Characteristic Curves Two-sided alternative: One-sided alternative: 11

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Type II Error and Choice of Sample Size Sample Size Formulas Two-sided alternative: 12

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Type II Error and Choice of Sample Size Sample Size Formulas One-sided alternative: 13

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Confidence Interval on a Difference in Means, Variances Known Definition 15

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Choice of Sample Size 18

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known One-Sided Confidence Bounds Upper Confidence Bound Lower Confidence Bound 19

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Hypotheses Tests for a Difference in Means, Variances Unknown We wish to test: Case 1: 20

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknow n Hypotheses Tests for a Difference in Means, Variances Unknown The pooled estimator of  2 : Case 1: 21

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Hypotheses Tests for a Difference in Means, Variances Unknown Case 1: 22

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Definition: The Two-Sample or Pooled t-Test * 23

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Minitab Output for Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Figure 10-2 Normal probability plot and comparative box plot for the catalyst yield data in Example (a) Normal probability plot, (b) Box plots. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 29

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Hypotheses Tests for a Difference in Means, Variances Unknown Case 2: is distributed approximately as t with degrees of freedom given by 30

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Hypotheses Tests for a Difference in Means, Variances Unknown Case 2: 31

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-6 (Continued) 33

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-6 (Continued) 34

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-6 (Continued) Figure 10-3 Normal probability plot of the arsenic concentration data from Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-6 (Continued) 36

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Type II Error and Choice of Sample Size Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Minitab Output for Example

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Confidence Interval on the Difference in Means, Variance Unknown Case 1: 39

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-8 Case 1: 40

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Case 1: Example 10-8 (Continued) 41

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Case 1: Example 10-8 (Continued) 42

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-8 (Continued) Case 1: 43

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Confidence Interval on the Difference in Means, Variance Unknown Case 2: 44

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. A special case of the two-sample t-tests of Section 10-3 occurs when the observations on the two populations of interest are collected in pairs. Each pair of observations, say (X 1j, X 2j ), is taken under homogeneous conditions, but these conditions may change from one pair to another. The test procedure consists of analyzing the differences between hardness readings on each specimen. 10-4: Paired t-Test 45

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. The Paired t-Test 10-4: Paired t-Test 46

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Paired t-Test 47

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Paired t-Test 48

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Paired t-Test 49

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Paired Versus Unpaired Comparisons 10-4: Paired t-Test 50

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. A Confidence Interval for  D 10-4: Paired t-Test Definition 51

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Paired t-Test 52

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Paired t-Test 53

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger The F Distribution 10-5 Inferences on the Variances of Two Normal Populations We wish to test the hypotheses: The development of a test procedure for these hypotheses requires a new probability distribution, the F distribution. 54

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger The F Distribution 10-5 Inferences on the Variances of Two Normal Populations 55

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger The F Distribution 10-5 Inferences on the Variances of Two Normal Populations 56

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger The F Distribution 10-5 Inferences on the Variances of Two Normal Populations The lower-tail percentage points f  -1,u, can be found as follows. 57

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Hypothesis Tests on the Ratio of Two Variances 10-5 Inferences on the Variances of Two Normal Populations 58

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Hypothesis Tests on the Ratio of Two Variances 10-5 Inferences on the Variances of Two Normal Populations 59

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example Inferences on the Variances of Two Normal Populations 60

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example Inferences on the Variances of Two Normal Populations 61

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example Inferences on the Variances of Two Normal Populations 62

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Type II Error and Choice of Sample Size 10-5 Inferences on the Variances of Two Normal Populations 63

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example Inferences on the Variances of Two Normal Populations 64

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Confidence Interval on the Ratio of Two Variances 10-5 Inferences on the Variances of Two Normal Populations 65

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example Inferences on the Variances of Two Normal Populations 66

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example Inferences on the Variances of Two Normal Populations 67

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Large-Sample Test on the Difference in Population Proportions 10-6: Inference on Two Population Proportions We wish to test the hypotheses : 68

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Large-Sample Test on the Difference in Population Proportions 10-6: Inference on Two Population Proportions The following test statistic is distributed approximately as standard normal and is the basis of the test : 69

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-6: Inference on Two Population Proportions Large-Sample Test on the Difference in Population Proportions 70

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Inference on Two Population Proportions 71

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Inference on Two Population Proportions 72

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Inference on Two Population Proportions 73

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Minitab Output for Example : Inference on Two Population Proportions 74

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Type II Error and Choice of Sample Size 10-6: Inference on Two Population Proportions 75

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Type II Error and Choice of Sample Size 10-6: Inference on Two Population Proportions 76

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Type II Error and Choice of Sample Size 10-6: Inference on Two Population Proportions 77

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger Confidence Interval on the Difference in the Population Proportions 10-6: Inference on Two Population Proportions 78

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Inference on Two Population Proportions 79

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Example : Inference on Two Population Proportions 80

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-7: Summary Table and Road Map for Inference Procedures for Two Samples Table

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10-7: Summary Table and Road Map for Inference Procedures for Two Samples Table 10-5 (Continued) 82

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Important Terms & Concepts of Chapter 10 Comparative experiments Confidence intervals on: Differences Ratios Critical region for a test statistic Identifying cause and effect Null and alternative hypotheses 1 & 2-sided alternative hypotheses Operating Characteristic (OC) curves Paired t-test Pooled t-test P-value Reference distribution for a test statistic Sample size determination for: Hypothesis tests Confidence intervals Statistical hypotheses Test statistic Wilcoxon rank-sum test 83