10
Statistical Inference
for Two Samples
CHAPTER OUTLINE
10-1 Inference on the Difference in Means of Two 10-5 Inference on the Variances of Two Normal
Normal Distributions, Variances Known
Populations
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
10-5.1 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
10-2 Inference on the Difference in Means of Two 10-6 Inference on Two Population Proportions
Normal Distributions, Variance Unknown
10-6.1 Large sample tests on the difference in
10-2.1 Hypothesis tests on the difference of means,
population proportions
variances unknown
10-6.2 Type II error and choice of sample size
10-2.2 Type II error and choice of sample size
10-6.3 Confidence interval on the difference in
10-2.3 Confidence interval on the difference in means,
population proportions
variance unknown
10-7 Summary Table and Roadmap for Inference
10-3 A Nonparametric Test on the Difference of
Two Means
10-4 Paired t-Tests
Procedures for Two Samples
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Learning Objectives for Chapter 10
After 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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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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© 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
8
© 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
9
© 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
10
© 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
Use 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:
13
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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
10-2.3 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
16
© 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
17
© 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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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
Case 1:
  
2
1
2
2
2
We wish to 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
10-3.1 Hypotheses Tests for a Difference in Means,
Variances Unknown
Case 1:
  
2
1
2
2
2
The pooled estimator of 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
10-3.1 Hypotheses Tests for a Difference in Means,
Variances Unknown
Case 1:
  
2
1
2
2
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
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
24
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10-3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Example 10-5
25
© 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
26
© 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
27
© 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.
10-3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
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.
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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
2
2
Case 2: 1  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
Case 2:
 
2
1
2
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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10-3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Example 10-6
(Continued)
33
© 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
37
© 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
38
© 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:
  
2
1
2
2
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
Case 1:
  
2
1
2
2
2
Example 10-8
40
© 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:
  
2
1
2
2
2
Example 10-8 (Continued)
41
© 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:
  
2
1
2
2
2
Example 10-8 (Continued)
42
© 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:
  
2
1
2
2
2
Example 10-8 (Continued)
43
© 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:
 
2
1
2
2
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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 10-10
47
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10-4: Paired t-Test
Example 10-10
48
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10-4: Paired t-Test
Example 10-10
49
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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 10-11
52
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10-4: Paired t-Test
Example 10-11
53
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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.
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10-5 Inferences on the Variances of Two Normal Populations
10-5.1 The F Distribution
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10-5 Inferences on the Variances of Two Normal Populations
10-5.1 The F Distribution
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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.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5 Inferences on the Variances of Two Normal Populations
10-5.2 Hypothesis Tests on the Ratio of Two
Variances
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5 Inferences on the Variances of Two Normal Populations
10-5.2 Hypothesis Tests on the Ratio of Two
Variances
59
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10-5 Inferences on the Variances of Two Normal Populations
Example 10-12
60
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10-5 Inferences on the Variances of Two Normal Populations
Example 10-12
61
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10-5 Inferences on the Variances of Two Normal Populations
Example 10-12
62
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5 Inferences on the Variances of Two Normal Populations
10-5.3 Type II Error and Choice of Sample Size
63
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10-5 Inferences on the Variances of Two Normal Populations
Example 10-13
64
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10-5 Inferences on the Variances of Two Normal Populations
10-5.4 Confidence Interval on the Ratio of Two
Variances
65
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-5 Inferences on the Variances of Two Normal Populations
Example 10-14
66
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10-5 Inferences on the Variances of Two Normal Populations
Example 10-14
67
© 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
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: 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:
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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
70
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6: Inference on Two Population Proportions
Example 10-15
71
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6: Inference on Two Population Proportions
Example 10-15
72
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6: Inference on Two Population Proportions
Example 10-15
73
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
10-6: Inference on Two Population Proportions
Minitab Output for Example 10-15
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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
10-6.2 Type II Error and Choice of Sample Size
75
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10-6: Inference on Two Population Proportions
10-6.2 Type II Error and Choice of Sample Size
76
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10-6: Inference on Two Population Proportions
10-6.2 Type II Error and Choice of Sample Size
77
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10-6: Inference on Two Population Proportions
10-6.3 Confidence Interval on the Difference in the
Population Proportions
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10-6: Inference on Two Population Proportions
Example 10-16
79
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10-6: Inference on Two Population Proportions
Example 10-16
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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)
82
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Important Terms & Concepts of Chapter 10
Comparative experiments
Paired t-test
Confidence intervals on:
Pooled t-test
• Differences
P-value
• Ratios
Reference distribution for a test
Critical region for a test statistic statistic
Sample size determination for:
Identifying cause and effect
Null and alternative hypotheses Hypothesis tests
Confidence intervals
1 & 2-sided alternative
Statistical hypotheses
hypotheses
Operating Characteristic (OC) Test statistic
curves
Wilcoxon rank-sum test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.