1 10 statistical inference for two samples 10-1 inference on the difference in means of two normal...

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10Statistical Inference for Two Samples

CHAPTER OUTLINE

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

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.

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10-1: Introduction

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

Figure 10-1 Two independent populations.

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Assumptions

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10-2.1 Hypothesis Tests for a Difference in Means, Variances Known

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Example 10-1

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Example 10-1

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Example 10-1

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10-2.2 Type II Error and Choice of Sample SizeUse of Operating Characteristic Curves

Two-sided alternative:

One-sided alternative:

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10-2.2 Type II Error and Choice of Sample Size

Sample Size Formulas

Two-sided alternative:

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Sample Size Formulas

One-sided alternative:

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Example 10-3

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10-2.3 Confidence Interval on a Difference in Means, Variances Known

Definition

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Example 10-4

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Example 10-4

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Choice of Sample Size

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One-Sided Confidence Bounds

Upper Confidence Bound

Lower Confidence Bound

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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

We wish to test:

Case 1: 22

221

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The pooled estimator of 2:

Case 1: 222

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Case 1: 222

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Definition: The Two-Sample or Pooled t-Test*

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Example 10-5

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Example 10-5

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Example 10-5

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Example 10-5

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Minitab Output for Example 10-5

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Figure 10-2 Normal probability plot and comparative box plot for the catalyst yield data in Example 10-5. (a) Normal probability plot, (b) Box plots.


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22

21 Case 2:

is distributed approximately as t with degrees of freedom given by

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22

21 Case 2:

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Example 10-6

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Example 10-6 (Continued)

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Example 10-6 (Continued)Figure 10-3 Normal probability plot of the arsenic concentration data from Example 10-6.

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Example 10-7

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10-3.3 Confidence Interval on the Difference in Means, Variance Unknown

Case 1: 222

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Example 10-8

Case 1: 222

21

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Case 1: 222

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Case 1: 222

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Case 1: 222

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10-3.3 Confidence Interval on the Difference in Means, Variance Unknown

Case 2: 22

21

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• 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.

10-4: Paired t-Test

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The Paired t-Test

10-4: Paired t-Test

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Example 10-10

10-4: Paired t-Test

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Example 10-10

10-4: Paired t-Test

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Example 10-10

10-4: Paired t-Test

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Paired Versus Unpaired Comparisons

10-4: Paired t-Test

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A Confidence Interval for D

10-4: Paired t-Test

Definition

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Example 10-11

10-4: Paired t-Test

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Example 10-11

10-4: Paired t-Test

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10-5.1 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.

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The lower-tail percentage points f-1,u, can be found as follows.

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10-5.2 Hypothesis Tests on the Ratio of Two Variances


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10-5.2 Hypothesis Tests on the Ratio of Two Variances


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Example 10-12


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Example 10-12


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Example 10-12


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Example 10-13


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10-5.4 Confidence Interval on the Ratio of Two Variances


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Example 10-14


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Example 10-14


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10-6.1 Large-Sample Test on the Difference in Population Proportions

10-6: Inference on Two Population Proportions

We wish to test the hypotheses:

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The following test statistic is distributed approximately as standard normal and is the basis of the test:

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Example 10-15


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Example 10-15


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Example 10-15


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10-6.3 Confidence Interval on the Difference in the Population Proportions


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Example 10-16


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Example 10-16


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10-7: Summary Table and Road Map for Inference Procedures for Two Samples

Table 10-5

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10-7: Summary Table and Road Map for Inference Procedures for Two Samples

Table 10-5 (Continued)

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Important Terms & Concepts of Chapter 10

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1 10 statistical inference for two samples 10-1 inference on the difference in means of two normal...

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