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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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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 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 t test1
10-4: Paired Distributions, Variances Unknownt-Test

The Paired t-Test


10 4 paired t test2
10-4: Paired Distributions, Variances Unknownt-Test

Example 10-10


10 4 paired t test3
10-4: Paired Distributions, Variances Unknownt-Test

Example 10-10


10 4 paired t test4
10-4: Paired Distributions, Variances Unknownt-Test

Example 10-10


10 4 paired t test5
10-4: Paired Distributions, Variances Unknownt-Test

Paired Versus Unpaired Comparisons


10 4 paired t test6
10-4: Paired Distributions, Variances Unknownt-Test

A Confidence Interval for D

Definition


10 4 paired t test7
10-4: Paired Distributions, Variances Unknownt-Test

Example 10-11


10 4 paired t test8
10-4: Paired Distributions, Variances Unknownt-Test

Example 10-11


10 5 inferences on the variances of two normal populations
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 populations1
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 populations2
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 populations3
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 populations4
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 populations5
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 populations6
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 populations7
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 populations8
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 populations9
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 populations10
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 populations11
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 populations12
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 populations13
10-5 Inferences on the Variances of Two Normal Populations Distributions, Variances Unknown

Example 10-14


10 6 inference on two population proportions
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 proportions1
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 proportions2
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 proportions3
10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Example 10-15


10 6 inference on two population proportions4
10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Example 10-15


10 6 inference on two population proportions5
10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Example 10-15


10 6 inference on two population proportions6
10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Minitab Output for Example 10-15


10 6 inference on two population proportions7
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 proportions8
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 proportions9
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 proportions10
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 proportions11
10-6: Inference on Two Population Proportions Distributions, Variances Unknown

Example 10-16


10 6 inference on two population proportions12
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 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 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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