Chapter 10: Statistical 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 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

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Chapter Learning Objectives After careful study of this chapter you should be able to: 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

Introduction

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Difference in Means or Variances of Two Normal Distributions

Figure 10-1: Two independent populations. 4

Difference in Means of Two Normal Distributions - Assumptions

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Difference in Means of Two Normal Distributions – Z statistic

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Difference in Means of Two Normal Distributions – Hypothesis Tests

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Example 10-1

Difference in Means of Two Normal Distributions – Example

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We may solve this problem using the seven-step procedure in Section 9-1.6: 1. Parameters of interest is the difference in the drying times, μ1 – μ2, and Δ0 = 0. 2. H0: μ1 – μ2 = 0 3. H1: μ1 – μ2 > 0 4. Test statistic is:

5. Reject H0 if the P - value is less than 0.05. 6. Compute:

7. Conclude: Since z0 =2.52, the P-value = Therefore we reject H0 at the 95% confidence level. This means that we

conclude adding the new ingredient to the paint reduces drying time.

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Type II Error Probability, β

Single-tailed:

Two-tailed:

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Sample Size Formulas for the Difference in Means of Two Normal Distributions

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Example 10-3

Choice of Sample Size – Example

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Definition:

C.I. on a Difference in Means, Variances Known

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Example 10-4

C.I. on a Difference in Means, Variances Known – Example

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Example 10-4 (continued)

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C.I. on a Difference in Means, Variances Known – Choice of Sample Size

If we assume equal sample sizes for the two populations, we can determine the sample size, n, required so that the error in estimating μ1 - μ2 will be less than E (the confidence interval half-width) at 100(1-α)% confidence.

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Upper Confidence Bound

Lower Confidence Bound

C.I. on a Difference in Means, Variances Known – One-Sided Tests

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• General methods in Section 10-2 to deal with the difference between means for two normal populations when the variances are unknown will not be covered. However, paired t-tests will be examined. These are a special case.

• In paired t-tests 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.

Paired t-tests

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The Paired t-Test

(10-24)

Example 10-10

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Example 10-10 (continued)

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Example 10-10 (continued)

Definition:

Paired t-test confidence interval for μD

(10-25)

Example 10-11

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The F Distribution

Inferences on the Variances of Two Normal Populations

Suppose 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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(10-26)

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The F Distribution (continued)

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(10-28)

(10-29)

The F Distribution (continued)

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The lower-tail percentage points f1-�,u,� can be found as follows.

The F Distribution (continued)

The percentage points for f�,�,u can be looked up from Table VI in the Appendix (p. 712).

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(10-30)

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If we plug the definition for the chi-squared distribution (Equation 8-17) in for W and Y, the following can be obtained.

The F Distribution (continued)

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The F Distribution (continued)

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Hypothesis Tests on the Ratio of Two Variances

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(10-31)

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Example 10-12

Hypothesis Tests on the Ratio of Two Variances - Example

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Fifteen

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We may solve this problem using the seven-step procedure in Section 9-1.6: 1. Parameters of interest are the variances of the oxide thicknesses, σ12 and

σ22. We assume that thicknesses are normal random variables. 2. H0: σ12 = σ22 3. H1: σ12 ≠ σ22 4. Test statistic is:

5. Reject H0 if f0 > f0.025, 15,15 =2.86 or if f0 < f0.975, 15,15 = 1/2.86 =0.35. 6. Compute:

7. Conclude: Since 0.35 < 0.85< 2.86, we cannot reject H0 at the 95% confidence level. This means there is no strong evidence either gas results is a smaller variance.

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C.I. on the Ratio of Two Variances

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(10-33)

Example 10-14

C.I. on the Ratio of Two Variances - Example

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Example 10-14 (continued)

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10-33

Example 10-14 (continued)

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10-30

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