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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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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

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

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 1 introduction

10-1: Introduction


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

Figure 10-1Two independent populations.


10 2 inference for a difference in means of two normal distributions variances known1

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 known2

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 known3

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 known4

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 known5

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 known6

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 known7

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 known8

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 known9

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 known10

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 known11

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 known12

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 known13

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 known14

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 known15

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: 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 unknow n

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 unknown1

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 unknown2

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 unknown3

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 unknown4

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 unknown5

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 unknown6

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 unknown7

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 unknown8

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 unknown9

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 unknown10

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 unknown11

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 unknown12

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 unknown13

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 unknown14

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 unknown15

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 unknown16

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 unknown17

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 unknown18

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 unknown19

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 unknown20

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 unknown21

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 unknown22

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 unknown23

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 t test

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.


10 4 paired t test1

10-4: Paired t-Test

The Paired t-Test


10 4 paired t test2

10-4: Paired t-Test

Example 10-10


10 4 paired t test3

10-4: Paired t-Test

Example 10-10


10 4 paired t test4

10-4: Paired t-Test

Example 10-10


10 4 paired t test5

10-4: Paired t-Test

Paired Versus Unpaired Comparisons


10 4 paired t test6

10-4: Paired t-Test

A Confidence Interval for D

Definition


10 4 paired t test7

10-4: Paired t-Test

Example 10-11


10 4 paired t test8

10-4: Paired t-Test

Example 10-11


10 5 inferences on the variances of two normal populations

10-5 Inferences on the Variances of Two Normal Populations

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 populations1

10-5 Inferences on the Variances of Two Normal Populations

10-5.1 The F Distribution


10 5 inferences on the variances of two normal populations2

10-5 Inferences on the Variances of Two Normal Populations

10-5.1 The F Distribution


10 5 inferences on the variances of two normal populations3

10-5 Inferences on the Variances of Two Normal Populations

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 populations4

10-5 Inferences on the Variances of Two Normal Populations

10-5.2 Hypothesis Tests on the Ratio of Two Variances


10 5 inferences on the variances of two normal populations5

10-5 Inferences on the Variances of Two Normal Populations

10-5.2 Hypothesis Tests on the Ratio of Two Variances


10 5 inferences on the variances of two normal populations6

10-5 Inferences on the Variances of Two Normal Populations

Example 10-12


10 5 inferences on the variances of two normal populations7

10-5 Inferences on the Variances of Two Normal Populations

Example 10-12


10 5 inferences on the variances of two normal populations8

10-5 Inferences on the Variances of Two Normal Populations

Example 10-12


10 5 inferences on the variances of two normal populations9

10-5 Inferences on the Variances of Two Normal Populations

10-5.3 Type II Error and Choice of Sample Size


10 5 inferences on the variances of two normal populations10

10-5 Inferences on the Variances of Two Normal Populations

Example 10-13


10 5 inferences on the variances of two normal populations11

10-5 Inferences on the Variances of Two Normal Populations

10-5.4 Confidence Interval on the Ratio of Two Variances


10 5 inferences on the variances of two normal populations12

10-5 Inferences on the Variances of Two Normal Populations

Example 10-14


10 5 inferences on the variances of two normal populations13

10-5 Inferences on the Variances of Two Normal Populations

Example 10-14


10 6 inference on two population proportions

10-6: Inference on Two Population Proportions

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

We wish to test the hypotheses:


10 6 inference on two population proportions1

10-6: Inference on Two Population Proportions

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 proportions2

10-6: Inference on Two Population Proportions

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


10 6 inference on two population proportions3

10-6: Inference on Two Population Proportions

Example 10-15


10 6 inference on two population proportions4

10-6: Inference on Two Population Proportions

Example 10-15


10 6 inference on two population proportions5

10-6: Inference on Two Population Proportions

Example 10-15


10 6 inference on two population proportions6

10-6: Inference on Two Population Proportions

Minitab Output for Example 10-15


10 6 inference on two population proportions7

10-6: Inference on Two Population Proportions

10-6.2 Type II Error and Choice of Sample Size


10 6 inference on two population proportions8

10-6: Inference on Two Population Proportions

10-6.2 Type II Error and Choice of Sample Size


10 6 inference on two population proportions9

10-6: Inference on Two Population Proportions

10-6.2 Type II Error and Choice of Sample Size


10 6 inference on two population proportions10

10-6: Inference on Two Population Proportions

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


10 6 inference on two population proportions11

10-6: Inference on Two Population Proportions

Example 10-16


10 6 inference on two population proportions12

10-6: Inference on Two Population Proportions

Example 10-16


10 7 summary table and road map for inference procedures for two samples

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 samples1

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

Table 10-5 (Continued)


Important terms concepts of chapter 10

Important Terms & Concepts of Chapter 10

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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