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Statistical Inference for Two Samples

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

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

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.

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

Figure 10-1Two independent populations.

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

Assumptions

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

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

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

Example 10-1

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

Example 10-1

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

Example 10-1

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:

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:

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:

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

Example 10-3

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

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

Example 10-4

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

Example 10-4

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

Choice of Sample Size

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

One-Sided Confidence Bounds

Upper Confidence Bound

Lower Confidence Bound

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:

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:

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:

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

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

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

Example 10-5

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

Example 10-5

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

Example 10-5

Example 10-5

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

Minitab Output for Example 10-5

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.

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

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:

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

Example 10-6

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

Example 10-6 (Continued)

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

Example 10-6 (Continued)

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.

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

Example 10-6 (Continued)

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

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

Minitab Output for Example 10-7

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:

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

Case 1:

Example 10-8

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

Case 1:

Example 10-8 (Continued)

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

Case 1:

Example 10-8 (Continued)

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

Case 1:

Example 10-8 (Continued)

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:

10-4: Paired Distributions, Variances Unknownt-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.

10-4: Paired Distributions, Variances Unknownt-Test

The Paired t-Test

10-4: Paired Distributions, Variances Unknownt-Test

Example 10-10

10-4: Paired Distributions, Variances Unknownt-Test

Example 10-10

10-4: Paired Distributions, Variances Unknownt-Test

Example 10-10

10-4: Paired Distributions, Variances Unknownt-Test

Paired Versus Unpaired Comparisons

10-4: Paired Distributions, Variances Unknownt-Test

Example 10-11

10-4: Paired Distributions, Variances Unknownt-Test

Example 10-11

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

10-5.1 The F Distribution

We wish to test the hypotheses:

- The development of a test procedure for these hypotheses requires a new probability distribution, the F distribution.

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

10-5.1 The F Distribution

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

10-5.1 The F Distribution

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

10-5.1 The F Distribution

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

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

10-5.2 Hypothesis Tests on the Ratio of Two Variances

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

10-5.2 Hypothesis Tests on the Ratio of Two Variances

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

Example 10-12

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

Example 10-12

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

Example 10-12

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

10-5.3 Type II Error and Choice of Sample Size

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

Example 10-13

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

10-5.4 Confidence Interval on the Ratio of Two Variances

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

Example 10-14

10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

Example 10-14

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

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

We wish to test the hypotheses:

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

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

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

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

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

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Example 10-15

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Example 10-15

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Example 10-15

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Minitab Output for Example 10-15

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

10-6.2 Type II Error and Choice of Sample Size

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

10-6.2 Type II Error and Choice of Sample Size

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

10-6.2 Type II Error and Choice of Sample Size

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

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

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Example 10-16

10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Example 10-16

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

Table 10-5

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

Table 10-5 (Continued)

Important Terms & Concepts of Chapter 10 for Two Samples

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

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