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Statistical Inference for Two Samples

10. Statistical Inference for Two Samples. CHAPTER OUTLINE. 10-1 Inference on the Difference in Means of Two Normal Distributions, Variances Known 10-1.1 Hypothesis tests on the difference of means, variances known 10-1.2 Type II error and choice of sample size

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Statistical Inference for Two Samples

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  1. 10 Statistical Inference for Two Samples CHAPTER OUTLINE 10-1 Inference on the Difference in Means of Two Normal Distributions, Variances Known 10-1.1 Hypothesis tests on the difference of means, variances known 10-1.2 Type II error and choice of sample size 10-1.3 Confidence interval on the difference in means, variance known 10-2 Inference on the Difference in Means of Two Normal Distributions, Variance Unknown 10-2.1 Hypothesis tests on the difference of means, variances unknown 10-2.2 Type II error and choice of sample size 10-2.3 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 10-5.1 F distributions 10-5.2 Hypothesis tests on the ratio of two variances 10-5.3 Type II error and choice of sample size 10-5.4 Confidence interval on the ratio of two variances 10-6 Inference on Two Population Proportions 10-6.1 Large sample tests on the difference in population proportions 10-6.2 Type II error and choice of sample size 10-6.3 Confidence interval on the difference in population proportions 10-7 Summary Table and Roadmap for Inference Procedures for Two Samples

  2. Learning Objectives for Chapter 10 After careful study of this chapter, you should be able to do the following: • Structure comparative experiments involving two samples as hypothesis tests. • Test hypotheses and construct confidence intervals on the difference in means of two normal distributions. • Test hypotheses and construct confidence intervals on the ratio of the variances or standard deviations of two normal distributions. • Test hypotheses and construct confidence intervals on the difference in two population proportions. • Use the P-value approach for making decisions in hypothesis tests. • Compute power, Type II error probability, and make sample size decisions for two-sample tests on means, variances & proportions. • Explain & use the relationship between confidence intervals and hypothesis tests.

  3. 10-1: Introduction

  4. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Figure 10-1Two independent populations.

  5. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Assumptions

  6. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known

  7. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known 10-2.1 Hypothesis Tests for a Difference in Means, Variances Known

  8. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example 10-1

  9. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example 10-1

  10. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example 10-1

  11. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known 10-2.2 Type II Error and Choice of Sample Size Use of Operating Characteristic Curves Two-sided alternative: One-sided alternative:

  12. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known 10-2.2 Type II Error andChoice of Sample Size Sample Size Formulas Two-sided alternative:

  13. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known 10-2.2 Type II Error and Choice of Sample Size Sample Size Formulas One-sided alternative:

  14. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example 10-3

  15. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known 10-2.3 Confidence Interval on a Difference in Means, Variances Known Definition

  16. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example 10-4

  17. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Example 10-4

  18. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known Choice of Sample Size

  19. 10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known One-Sided Confidence Bounds Upper Confidence Bound Lower Confidence Bound

  20. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown Case 1: We wish to test:

  21. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown Case 1: The pooled estimator of 2:

  22. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown Case 1:

  23. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Definition: The Two-Sample or Pooled t-Test*

  24. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-5

  25. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-5

  26. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-5

  27. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-5

  28. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Minitab Output for Example 10-5

  29. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Figure 10-2Normal probability plot and comparative box plot for the catalyst yield data in Example 10-5. (a) Normal probability plot, (b) Box plots.

  30. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown Case 2: is distributed approximately as t with degrees of freedom given by

  31. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 10-3.1 Hypotheses Tests for a Difference in Means, Variances Unknown Case 2:

  32. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-6

  33. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-6 (Continued)

  34. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-6 (Continued)

  35. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-6 (Continued) Figure 10-3Normal probability plot of the arsenic concentration data from Example 10-6.

  36. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Example 10-6 (Continued)

  37. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 10-3.2 Type II Error and Choice of Sample Size Example 10-7

  38. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Minitab Output for Example 10-7

  39. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 10-3.3 Confidence Interval on the Difference in Means, Variance Unknown Case 1:

  40. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Case 1: Example 10-8

  41. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Case 1: Example 10-8 (Continued)

  42. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Case 1: Example 10-8 (Continued)

  43. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown Case 1: Example 10-8 (Continued)

  44. 10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown 10-3.3 Confidence Interval on the Difference in Means, Variance Unknown Case 2:

  45. 10-4: Paired t-Test • 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 (X1j, X2j), 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.

  46. 10-4: Paired t-Test The Paired t-Test

  47. 10-4: Paired t-Test Example 10-10

  48. 10-4: Paired t-Test Example 10-10

  49. 10-4: Paired t-Test Example 10-10

  50. 10-4: Paired t-Test Paired Versus Unpaired Comparisons

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