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7/27/2019 Chapter 10 Two Sample Inferences
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1
10Statistical Inference
for Two Samples
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 Unknown10-2.1 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
CHAPTER OUTLINE
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John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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 inmeans of two normal distributions.
3. Test hypotheses and construct confidence intervals on the ratio of thevariances or standard deviations of two normal distributions.
4. Test hypotheses and construct confidence intervals on the difference intwo 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 decisionsfor two-sample tests on means, variances & proportions.
7. Explain & use the relationship between confidence intervals andhypothesis tests.
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10-2: Inference for a Difference in Means of Two Normal
Distributions, Variances Known
Figure 10-1Two independent populations.
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10-2: Inference for a Difference in Means of Two Normal
Distributions, Variances Known
Assumptions
4
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10-2: Inference for a Difference in Means of Two Normal
Distributions, Variances Known
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10-2.1 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
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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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:
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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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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
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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
17
f f ff f
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10-2: Inference for a Difference in Means of Two Normal
Distributions, Variances Known
One-Sided Confidence BoundsUpper Confidence Bound
Lower Confidence Bound
18
f f iff i f l
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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
2
2
1
19
10 3 I f f Diff i M f T N l
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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 of2:
Case 1: 222
21
20
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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
21
21
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10-3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Definition: The Two-Sample or Pooledt
-Test
*
22
10 3 I f f Diff i M f T N l
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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 I f f Diff i M f T N l
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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
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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
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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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10 3: Inference for a Difference in Means of Two Normal
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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 Unknown
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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 Unknown2
2
2
1Case 2:
is distributed approximately as twith degrees of freedom
given by
29
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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
2
1 Case 2:
30
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10-3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Example 10-6
31
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10-3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Example 10-6
(Continued)
32
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10-3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Example 10-6 (Continued)
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10-3: Inference for a Difference in Means of Two Normal
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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.
34
10-3: Inference for a Difference in Means of Two Normal
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10-3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Example 10-6 (Continued)
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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
36
10-3: Inference for a Difference in Means of Two Normal
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10 3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Minitab Output for Example 10-7
37
10-3: Inference for a Difference in Means of Two Normal
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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 1:22
2
2
1
38
10-3: Inference for a Difference in Means of Two Normal
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10 3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Example 10-8
Case 1:22
2
2
1
39
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10 3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Case 1:22
2
2
1
Example 10-8 (Continued)
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10 3: Inference for a Difference in Means of Two Normal
Distributions, Variances Unknown
Case 1:22
2
2
1
Example 10-8 (Continued)
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0 3: Inference for a ifference in Means of Two Normal
Distributions, Variances Unknown
Example 10-8 (Continued)
Case 1:22
2
2
1
42
10-3: Inference for a Difference in Means of Two Normal
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Distributions, Variances Unknown
10-3.3 Confidence Interval on the Difference in Means,
Variance Unknown
Case 2:2
2
2
1
43
10 4 Paired t Test
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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 thedifferences between hardness readings on each
specimen.
10-4: Paired t-Test
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10 4: Paired t Test
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The Paired t -Test
10-4: Paired t-Test
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Example 10-10
10-4: Paired t-Test
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10 4: Paired t Test
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Example 10-10
10-4: Paired t-Test
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10 4: Paired t Test
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Example 10-10
10-4: Paired t-Test
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10 4: Paired t Test
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Paired Versus Unpaired Comparisons
10-4: Paired t-Test
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10 4: Paired t Test
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A Confidence Interval forD
10-4: Paired t-Test
Definition
50
10 4: Paired t Test
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Example 10-11
10-4: Paired t-Test
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Example 10-11
10-4: Paired t-Test
52
10-5 Inferences on the Variances of Two Normal Populations
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10-5.1 The FDistribution
10-5 Inferences on the Variances of Two Normal Populations
We wish to test the hypotheses:
The development of a test procedure for thesehypotheses requires a new probability distribution, the
Fdistribution.
53
10-5 Inferences on the Variances of Two Normal Populations
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10-5.1 The FDistribution
10-5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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10-5.1 The FDistribution
10-5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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10-5.1 The FDistribution
10-5 Inferences on the Variances of Two Normal Populations
The lower-tail percentage points f-1,u, can be found as follows.
56
10-5 Inferences on the Variances of Two Normal Populations
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10-5.2 Hypothesis Tests on the Ratio of TwoVariances
10-5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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10-5.2 Hypothesis Tests on the Ratio of TwoVariances
10 5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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Example 10-12
10 5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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Example 10-12
10 5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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Example 10-12
10 5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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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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10-5 Inferences on the Variances of Two Normal Populations
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Example 10-13
10 5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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10-5.4 Confidence Interval on the Ratio of TwoVariances
10 5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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Example 10-14
10 5 Inferences on the Variances of Two Normal Populations
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10-5 Inferences on the Variances of Two Normal Populations
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Example 10-14
p
66
10-6: Inference on Two Population Proportions
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10-6.1 Large-Sample Test on the Difference inPopulation Proportions
0 6: Inference on Two Population Proportions
We wish to test the hypotheses:
67
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10-6.1 Large-Sample Test on the Difference in PopulationProportions
p p
The following test statistic is distributed
approximately as standard normal and is thebasis of the test:
68
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p p
10-6.1 Large-Sample Test on the Difference in Population
Proportions
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Example 10-15
p p
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Example 10-15
p p
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Example 10-15
p p
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Minitab Output for Example 10-15
p p
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10-6.2 Type II Error and Choice of Sample Size
p p
74
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10-6.2 Type II Error and Choice of Sample Size
p p
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10-6: Inference on Two Population Proportions
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10-6.2 Type II Error and Choice of Sample Size
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10-6: Inference on Two Population Proportions
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10-6.3 Confidence Interval on the Difference in the
Population Proportions
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10-6: Inference on Two Population Proportions
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Example 10-16
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10-6: Inference on Two Population Proportions
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Example 10-16
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10-7: Summary Table and Road Map for Inference Procedures
for Two Samples
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for Two SamplesTable 10-5
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10-7: Summary Table and Road Map for Inference Proceduresfor Two Samples
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for Two Samples
Table 10-5 (Continued)
81
Important Terms & Concepts of Chapter 10
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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-testP-value
Reference distribution for a test
statistic
Sample size determination for:Hypothesis tests
Confidence intervals
Statistical hypotheses
Test statistic
Wilcoxon rank-sum test