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10-1 Introduction

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

Figure 10-1 Two independent populations.

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

Assumptions

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

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

Hypothesis Tests for a Difference in Means, Variances Known

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

Example 10-1

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

Example 10-1

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

Example 10-1

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

Choice of Sample Size Use of Operating Characteristic Curves Two-sided alternative: One-sided alternative:

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

Choice of Sample Size Sample Size Formulas Two-sided alternative:

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

Choice of Sample Size Sample Size Formulas One-sided alternative:

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

Example 10-3

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

Identifying Cause and Effect When statistical significance is observed in a randomized experiment, the experimenter can be confident in the conclusion that it was the difference in treatments that resulted in the difference in response. That is, we can be confident that a cause-and-effect relationship has been found.

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

Confidence Interval on a Difference in Means, Variances Known Definition

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

Example 10-4

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

Example 10-4

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

Choice of Sample Size

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

One-Sided Confidence Bounds Upper Confidence Bound Lower Confidence Bound

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

Hypotheses Tests for a Difference in Means, Variances Unknown Case 1: We wish to test:

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

Hypotheses Tests for a Difference in Means, Variances Unknown Case 1: The pooled estimator of 2:

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

Hypotheses Tests for a Difference in Means, Variances Unknown Case 1:

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

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

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

Example 10-5

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

Example 10-5

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

Example 10-5

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

Example 10-5

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

Minitab Output for Example 10-5

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Figure 10-2 Normal probability plot and comparative box plot for the catalyst yield data in Example (a) Normal probability plot, (b) Box plots.

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

Hypotheses Tests for a Difference in Means, Variances Unknown Case 2: is distributed approximately as t with degrees of freedom given by

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

Hypotheses Tests for a Difference in Means, Variances Unknown Case 2:

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

Example 10-6

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Example 10-6 Figure 10-3 Normal probability plot of the arsenic concentration data from Example 10-6.

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Example 10-6

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Example 10-6

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

Choice of Sample Size Two-sided alternative:

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

Example 10-7

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

Minitab Output for Example 10-7

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**10-3.4 Confidence Interval on the Difference in Means**

Case 1:

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**10-3.4 Confidence Interval on the Difference in Means**

Case 1: Example 10-8

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**10-3.4 Confidence Interval on the Difference in Means**

Case 1: Example 10-8

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**10-3.4 Confidence Interval on the Difference in Means**

Case 1: Example 10-8

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**10-3.4 Confidence Interval on the Difference in Means**

Case 1: Example 10-8

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**10-3.4 Confidence Interval on the Difference in Means**

Case 2: Definition

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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.

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10-4 Paired t-Test The Paired t-Test

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10-4 Paired t-Test Example 10-9

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10-4 Paired t-Test Example 10-9

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10-4 Paired t-Test Example 10-9

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10-4 Paired t-Test Paired Versus Unpaired Comparisons

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10-4 Paired t-Test A Confidence Interval for D Definition

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10-4 Paired t-Test Example 10-10

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10-4 Paired t-Test Example 10-10

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

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.

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

The F Distribution

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

The F Distribution

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

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

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

Hypothesis Tests on the Ratio of Two Variances

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

Hypothesis Tests on the Ratio of Two Variances

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

Example 10-11

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

Example 10-11

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

Example 10-11

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

-Error and Choice of Sample Size

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

Example 10-12

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

Confidence Interval on the Ratio of Two Variances

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

Example 10-13

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

Example 10-13

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

Example 10-13

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**10-6 Inference on Two Population Proportions**

Large-Sample Test for H0: p1 = p2 We wish to test the hypotheses:

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**10-6 Inference on Two Population Proportions**

Large-Sample Test for H0: p1 = p2 The following test statistic is distributed approximately as standard normal and is the basis of the test:

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**10-6 Inference on Two Population Proportions**

Large-Sample Test for H0: p1 = p2

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**10-6 Inference on Two Population Proportions**

Example 10-14

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**10-6 Inference on Two Population Proportions**

Example 10-14

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**10-6 Inference on Two Population Proportions**

Example 10-14

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**10-6 Inference on Two Population Proportions**

Minitab Output for Example 10-14

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**10-6 Inference on Two Population Proportions**

-Error and Choice of Sample Size

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**10-6 Inference on Two Population Proportions**

-Error and Choice of Sample Size

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**10-6 Inference on Two Population Proportions**

-Error and Choice of Sample Size

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**10-6 Inference on Two Population Proportions**

Confidence Interval for p1 – p2

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**10-6 Inference on Two Population Proportions**

Example 10-15

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**10-6 Inference on Two Population Proportions**

Example 10-15

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