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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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slide1

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 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 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 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 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 populations11
10-5 Inferences on the Variances of Two Normal Populations

10-5.4 Confidence Interval on the Ratio of Two Variances

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 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 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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