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LESSON 19: CONFIDENCE INTERVALS FOR
THE DIFFERENCE BETWEEN MEANS
Outline
• Confidence intervals for the difference between
means
– Large sample
– Small sample equal variance
– Small sample unequal variance
– Matched pairs
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DIFFERENCE BETWEEN MEANS
• Let,
 A  Mean of population A
 B  Mean of population B
 A  Standard deviation of population A
 B  Standard deviation of population B
nA  Size of sample drawn from population A
nB  Size of sample drawn from population B
X A  Mean of the sample drawn from population A
X A  Mean of the sample drawn from population B
 A  Sample standard deviation, A
 B  Sample standard deviation, B
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DIFFERENCE BETWEEN MEANS
• We discuss the computation of confidence interval of
A  B
• Define the random variable,
D XA XB
• Then,
D   A  B
 
2
D
 A2
nA

 B2
nB
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LARGE SAMPLE
• If each of the two sample sizes is greater than 30, the
confidence interval estimate of
A  B
is computed as follows:
D XAXB
s A2 s B2
sD 

n A nB
 A   B  D  z / 2 sD
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DIFFERENCE BETWEEN MEANS
Example 1: An engineering society wishes to determine by how
much, if at all, the mean outcome of practicing mechanical
engineers exceeds that of electrical engineers. Two
independent random samples provided the following data:
Mechanical Engineers Electrical Engineers
n A  75
nB  60
X A  $82,200
X B  $81,400
s A  $8,400
sB  $8,800
Construct a 90% confidence interval estimate for the
difference in population means.,
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SMALL SAMPLE AND EQUAL VARIANCE
• If any of the two sample is less than 30, and both samples
have an equal variance, the confidence interval estimate of
A  B
is computed as follows:
D XAXB
sD 
n A  1s A2  nB  1sB2
n A  nB  2
1
1

n A nB
 A   B  D  t / 2 sD
df  n A  nB  2
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SMALL SAMPLE AND EQUAL VARIANCE
Example 2: Construct a 99% confidence interval estimate of the
difference between the MTBFs of two types of power cells,
assuming that the following results have been obtained for
independent samples. Assume equal population variances.
Cell A
Cell B
1
540 days
500 days
2
450
435
3
620
645
4
730
610
5
270
370
6
550
475
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SMALL SAMPLE AND UNEQUAL VARIANCE
• If any of the two sample is less than 30, and the samples
have unequal variances, the confidence interval estimate of
A  B
is computed as follows:
D XAXB
s A2 sB2
sD 

n A nB
 A   B  D  t / 2 sD
df 
s
2
A

s
2
A

2
2
B B
/ nA  s n


/ n A / n A  1  sB2 nB / nB  1
2
2
(round to nearest integer)
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SMALL SAMPLE AND UNEQUAL VARIANCE
Example 3: Redo Example 2 assuming unequal population
variance.
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MATCHED-PAIR SAMPLES
• So far, we discussed procedures that compares means of
two samples drawn independently of one another.
• However, matched-pair samples require that for every item
drawn from one population there be an item drawn from the
other population.
• For example, suppose that it is required to compare
performance of two makes of engines A and B. A random
sample of 25 engines are selected from type A engines.
Each type A engine is paired with a type B engine, also
randomly selected. Each match is based on age and size of
engine. Such a sample is a matched-pair sample.
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MATCHED-PAIR SAMPLES
• Letting i denote the i-th pair, with
and
representing
the sample observations for that pair from the respective
group, the confidence interval estimate of
A  B
is computed as follows:
d

d
i
n
sD 
 d
i
d

2
n 1
 A   B  d  t / 2
sD
, df  n  1
n
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MATCHED-PAIR SAMPLES
Example 4: Suppose that the observations in Example 2 have
been matched according to type of application, environmental
condition, and power demands. The pairs are as listed.
Construct a 99% confidence interval estimate for the
difference in MTBFs between the two types of power cell.
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READING AND EXERCISES
Lesson 19
Reading:
Section 10-5, pp. 319-330
Exercises:
10-37, 10-38, 10-39, 10-42
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