Chapter 10 Two Sample Inferences

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10Statistical Inference

for Two Samples

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 Unknown10-2.1 Hypothesis tests on the difference of means,

variances unknown


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


10-5.4 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


10-6.3 Confidence interval on the difference in

population proportions

10-7 Summary Table and Roadmap for Inference

Procedures for Two Samples

CHAPTER OUTLINE


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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 inmeans of two normal distributions.

3. Test hypotheses and construct confidence intervals on the ratio of thevariances or standard deviations of two normal distributions.

4. Test hypotheses and construct confidence intervals on the difference intwo 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 decisionsfor two-sample tests on means, variances & proportions.

7. Explain & use the relationship between confidence intervals andhypothesis tests.

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10-2: Inference for a Difference in Means of Two Normal

Distributions, Variances Known

Figure 10-1Two 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 Size

Use of Operating Characteristic Curves

Two-sided alternative:

One-sided alternative:

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10-2.2 Type II Error andChoice 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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f f ff f





One-Sided Confidence BoundsUpper Confidence Bound

Lower Confidence Bound

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f f iff i f l




Distributions, Variances Unknown

10-3.1 Hypotheses Tests for a Difference in Means,

Variances Unknown

We wish to test:

Case 1:

22

2

2

1

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10 3 I f f Diff i M f T N l






Variances Unknown

The pooled estimator of2:

Case 1: 222

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Variances Unknown

Case 1: 222

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Definition: The Two-Sample or Pooledt

-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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10 3: Inference for a Difference in Means of Two Normal


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Example 10-5

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Minitab Output for Example 10-5

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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.



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Variances Unknown2

2

2

1Case 2:

is distributed approximately as twith degrees of freedom

given by

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Variances Unknown

2

2

2

1 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)

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Example 10-6 (Continued)Figure 10-3Normal 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 1:22

2

2

1

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Example 10-8

Case 1:22

2

2

1

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Case 1:22

2

2

1


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Case 1:22

2

2

1


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0 3: Inference for a ifference in Means of Two Normal



Case 1:22

2

2

1

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10-3.3 Confidence Interval on the Difference in Means,

Variance Unknown

Case 2:2

2

2

1

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10 4 Paired t Test


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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 thedifferences between hardness readings on each

specimen.

10-4: Paired t-Test

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10 4: Paired t Test


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The Paired t -Test

10-4: Paired t-Test

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10 4: Paired t Test


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Example 10-10

10-4: Paired t-Test

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10 4: Paired t Test


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Example 10-10

10-4: Paired t-Test

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10 4: Paired t Test


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Example 10-10

10-4: Paired t-Test

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10 4: Paired t Test


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Paired Versus Unpaired Comparisons

10-4: Paired t-Test

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10 4: Paired t Test


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A Confidence Interval forD

10-4: Paired t-Test

Definition

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10 4: Paired t Test


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Example 10-11

10-4: Paired t-Test

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10 4: Paired t Test


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Example 10-11

10-4: Paired t-Test

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10-5 Inferences on the Variances of Two Normal Populations


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10-5.1 The FDistribution


We wish to test the hypotheses:

The development of a test procedure for thesehypotheses requires a new probability distribution, the

Fdistribution.

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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 TwoVariances


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10-5.2 Hypothesis Tests on the Ratio of TwoVariances

10 5 Inferences on the Variances of Two Normal Populations

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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 TwoVariances


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Example 10-14


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Example 10-14

p

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10-6: Inference on Two Population Proportions


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10-6.1 Large-Sample Test on the Difference inPopulation Proportions

0 6: Inference on Two Population Proportions

We wish to test the hypotheses:

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10-6.1 Large-Sample Test on the Difference in PopulationProportions

p p

The following test statistic is distributed

approximately as standard normal and is thebasis of the test:

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p p

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

Proportions

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Example 10-15

p p

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Example 10-15

p p

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Example 10-15

p p

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p p

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p p

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p p

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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


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for Two SamplesTable 10-5

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


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for Two Samples

Table 10-5 (Continued)

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Important Terms & Concepts of Chapter 10


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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-testP-value

Reference distribution for a test

statistic

Sample size determination for:Hypothesis tests

Confidence intervals

Statistical hypotheses

Test statistic

Wilcoxon rank-sum test

Chapter 10 Two Sample Inferences

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