Applied Statistics and Probability for
Engineers
Sixth Edition
Douglas C. Montgomery
George C. Runger
Chapter 10
Statistical Inference for Two Samples
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Statistical Inference
for Two Samples
10
CHAPTER OUTLINE
10-1 Inference on the Difference in Means of
Two Normal Distributions, Variances Known
10-1.1 Hypothesis tests on the difference in 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 in 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
in 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
2
Chapter 10 Title and Outline
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Wilcoxon rank-sum test
• X1 and X2 with means µ1 and µ2, but we are unwilling to
assume that they are (approximately) normal.
• Want to test:
• Let X11, X12,...,X1n1 and X21, X22,...,X2n2 be two
independent random samples of sizes n1 ≤ n2
3
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Procedure
• Arrange all n1 + n2 observations in ascending order of
magnitude and assign ranks to them.
– If two or more observations are tied (identical), use
the mean of the ranks that would have been assigned
if the observations differed.
• Let W1 be the sum of the ranks in the smaller sample (1),
and define W2 to be the sum of the ranks in the other
sample. Then,
4
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Procedure
• Get the critical value (wα) from Appendex
Table X
• Reject if either w1 or w2 is less than wα
– For H1 :µ1 <µ2 , reject H0 if w1≤ wα
– for H1 :µ1 >µ2 , reject H0 if w2 ≤ wα
5
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Table X
6
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Table X
7
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Example
•
The mean axial stress in tensile members used in an aircraft structure is
being studied. Two alloys are being investigated. Alloy 1 is a traditional
material, and alloy 2 is a new aluminum-lithium alloy that is much lighter
than the standard material. Ten specimens of each alloy type are tested,
and the axial stress is measured. The sample data are assembled in
Table 10-2. Using α= 0.05, we wish to test the hypothesis that the
means of the two stress distributions are identical.
8
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Example
1.
Parameter of interest: The parameters of interest are the means of the
two distributions of axial stress.
2.
Null hypothesis: H0: µ = µ .
1
2
3.
Alternative hypothesis: H1: µ ≠ µ
1
2
4.
α=0.05
5.
Test statistic: We will use the Wilcoxon rank-sum test statistic in
Equation 10-21.
9
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.
Example
6.
7.
8.
Rejection criteria :
–
If either w1 or w2 is less than or equal to w
= 78, we will reject H0: µ1 = µ2.
0.05
Computations:
Since both w and w > 78
1
2
We cannot reject H .
0
10
Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.