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10Statistical Inference for Two Samples
CHAPTER OUTLINE
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Learning Objectives for Chapter 10After careful study of this chapter, you should be able to do the
following: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.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-1: Introduction
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
Figure 10-1 Two independent populations.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
Assumptions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
10-2.2 Type II Error and Choice of Sample SizeUse of Operating Characteristic Curves
Two-sided alternative:
One-sided alternative:
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
Two-sided alternative:
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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:
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-3
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-4
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
Example 10-4
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-2: Inference for a Difference in Means of Two Normal Distributions, Variances Known
Choice of Sample Size
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
We wish to test:
Case 1: 22
221
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
The pooled estimator of 2:
Case 1: 222
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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: 222
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-5
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-5
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-5
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-5
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Minitab Output for Example 10-5
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Figure 10-2 Normal 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 Unknown
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
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21 Case 2:
is distributed approximately as t with degrees of freedom given by
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
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21 Case 2:
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-6
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-6 (Continued)
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-6 (Continued)
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-6 (Continued)Figure 10-3 Normal probability plot of the arsenic concentration data from Example 10-6.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-6 (Continued)
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Minitab Output for Example 10-7
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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: 222
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-8
Case 1: 222
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Case 1: 222
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Example 10-8 (Continued)
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Case 1: 222
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Example 10-8 (Continued)
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-3: Inference for a Difference in Means of Two Normal Distributions, Variances Unknown
Example 10-8 (Continued)
Case 1: 222
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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: 22
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
• 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-Test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
The Paired t-Test
10-4: Paired t-Test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-10
10-4: Paired t-Test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-10
10-4: Paired t-Test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-10
10-4: Paired t-Test
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Paired Versus Unpaired Comparisons
10-4: Paired t-Test
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A Confidence Interval for D
10-4: Paired t-Test
Definition
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-11
10-4: Paired t-Test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-11
10-4: Paired t-Test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5.1 The F Distribution
10-5 Inferences on the Variances of Two Normal Populations
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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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5.1 The F Distribution
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5.1 The F Distribution
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5.1 The F Distribution
10-5 Inferences on the Variances of Two Normal Populations
The lower-tail percentage points f-1,u, can be found as follows.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5.2 Hypothesis Tests on the Ratio of Two Variances
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5.2 Hypothesis Tests on the Ratio of Two Variances
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-12
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-12
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-12
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5.3 Type II Error and Choice of Sample Size
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-13
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5.4 Confidence Interval on the Ratio of Two Variances
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-14
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-14
10-5 Inferences on the Variances of Two Normal Populations
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6.1 Large-Sample Test on the Difference in Population Proportions
10-6: Inference on Two Population Proportions
We wish to test the hypotheses:
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6.1 Large-Sample Test on the Difference in Population Proportions
10-6: Inference on Two Population Proportions
The following test statistic is distributed approximately as standard normal and is the basis of the test:
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6: Inference on Two Population Proportions
10-6.1 Large-Sample Test on the Difference in Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-15
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-15
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-15
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Minitab Output for Example 10-15
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6.2 Type II Error and Choice of Sample Size
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6.2 Type II Error and Choice of Sample Size
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6.2 Type II Error and Choice of Sample Size
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6.3 Confidence Interval on the Difference in the Population Proportions
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-16
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 10-16
10-6: Inference on Two Population Proportions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-7: Summary Table and Road Map for Inference Procedures for Two Samples
Table 10-5
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10-7: Summary Table and Road Map for Inference Procedures for Two Samples
Table 10-5 (Continued)
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Important Terms & Concepts of Chapter 10
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