Transcript INFERENCE: TWO POPULATIONS

Chapter 10
Statistical Inference About Means and
Proportions With Two Populations
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Estimation of the Difference between the Means of
Two Populations: Independent Samples
Hypothesis Tests about the Difference between the
Means of Two Populations: Independent Samples
Inference about the Difference between the Means of
Two Populations: Matched Samples
Inference about the Difference between the
Proportions of Two Populations
Slide 1
Estimation of the Difference Between the Means
of Two Populations: Independent Samples
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Point Estimator of the Difference between the Means
of Two Populations
Sampling Distribution x1  x2
Interval Estimate of Large-Sample Case
Interval Estimate of Small-Sample Case
Slide 2
Point Estimator of the Difference Between
the Means of Two Populations
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Let 1 equal the mean of population 1 and 2 equal
the mean of population 2.
The difference between the two population means is
1 - 2.
To estimate 1 - 2, we will select a simple random
sample of size n1 from population 1 and a simple
random sample of size n2 from population 2.
Let x1 equal the mean of sample 1 and x2 equal the
mean of sample 2.
The point estimator of the difference between the
means of the populations 1 and 2 is x1  x2 .
Slide 3
Sampling Distribution of x1  x2

The sampling distribution of x1  x2 has the following
properties.
E( x1  x2 )  1  2
Expected Value:
Standard Deviation: 
x1  x2 
where
 12
n1

 22
n2
1 = standard deviation of population 1
2 = standard deviation of population 2
n1 = sample size from population
n2 = sample size from population 2
Slide 4
Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and n2 > 30)
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Interval Estimate with 1 and 2 Known
where
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x1  x2  z / 2 x1  x2
1 -  is the confidence coefficient
Interval Estimate with 1 and 2 Unknown
where
x1  x2  z / 2 sx1  x2
s x1  x2 
s12
s22

n1
n2
Slide 5
Example: Par, Inc.
Par, Inc. is a manufacturer of golf equipment. Par
has developed a new golf ball that has been designed to
provide “extra distance.” In a test of driving distance
using a mechanical driving device, a sample of Par golf
balls was compared with a sample of golf balls made by
Rap, Ltd., a competitor. The sample data is below.
Sample #1
Sample #2
Par, Inc.
Rap, Ltd.
Sample Size
n1 = 120 balls
n2 = 80 balls
x1 = 235 yards
x2 = 218 yards
Mean
Standard Deviation
s1 = 15 yards
s2 = 20 yards
Slide 6
Example: Par, Inc.

Point Estimate of the Difference Between Two
Population Means
1 = mean distance for the population of
Par, Inc. golf balls
2 = mean distance for the population of
Rap, Ltd. golf balls
Point estimate of 1 - 2 = x1  x2 = 235 - 218 = 17 yards.
Slide 7
Example: Par, Inc.
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95% Confidence Interval Estimate of the Difference
Between Two Population Means: Large-Sample Case,
1 and 2 Unknown
Substituting the sample standard deviations for the
population standard deviation:
x1  x2  z / 2
 12  22
(15)2 (20)2

 17  1.96

n1 n2
120
80
= 17 + 5.14 or 11.86 yards to 22.14 yards.
We are 95% confident that the difference between the
mean driving distances of Par, Inc. balls and Rap, Ltd.
balls lies in the interval of 11.86 to 22.14 yards.
Slide 8
Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30 and/or n2 < 30)
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Interval Estimate with  2 Known
where
x1  x2  z / 2 x1 x2
1
1
 x1  x2   ( 
)
n1
n2
Interval Estimate with  2 Unknown
2

where
sx1  x2
1
2 1
 s (  )
n1 n2
x1  x2  t / 2 sx1  x2
2
2
(
n

1
)
s

(
n

1
)
s
2
1
2
2
s  1
n1  n2  2
Slide 9
Example: Specific Motors
Specific Motors of Detroit has developed a new
automobile known as the M car. 12 M cars and 8 J cars
(from Japan) were road tested to compare miles-pergallon (mpg) performance. The sample data is below.
Sample Size
Mean
Standard Deviation
Sample #1
M Cars
n1 = 12 cars
x1 = 29.8 mpg
s1 = 2.56 mpg
Sample #2
J Cars
n2 = 8 cars
x2 = 27.3 mpg
s2 = 1.81 mpg
Slide 10
Example: Specific Motors
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Point Estimate of the Difference Between Two
Population Means
1 = mean miles-per-gallon for the population of
M cars
2 = mean miles-per-gallon for the population of
J cars
Point estimate of 1 - 2 = x1  x2 = 29.8 - 27.3 = 2.5 mpg.
Slide 11
Example: Specific Motors
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95% Confidence Interval Estimate of the Difference
Between Two Population Means: Small-Sample Case
For the small-sample case we will make the following
assumption.
1. The miles per gallon rating must be normally
distributed for both the M car and the J car.
2. The variance in the miles per gallon rating must
be the same for both the M car and the J car.
Using the t distribution with n1 + n2 - 2 = 18 degrees
of freedom, the appropriate t value is t.025 = 2.101.
We will use a weighted average of the two sample
variances as the pooled estimator of  2.
Slide 12
Example: Specific Motors
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95% Confidence Interval Estimate of the Difference
Between Two Population Means: Small-Sample Case
2
2
2
2
(
n

1
)
s

(
n

1
)
s
11
(
2
.
56
)

7
(
1
.
81
)
1
2
2
s2  1

 5.28
n1  n2  2
12  8  2
x1  x2  t.025
s2 (
1
1
1
1

)  2.5  2.101 5.28(
 )
n1 n2
12 8
= 2.5 + 2.2 or .3 to 4.7 miles per gallon.
We are 95% confident that the difference between the
mean mpg ratings of the two car types is from .3 to 4.7
mpg (with the M car having the higher mpg).
Slide 13
Hypothesis Tests About the Difference
Between the Means of Two Populations:
Independent Samples
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Hypothesis Forms:
H0: 1 - 2 < 0
H0: 1 - 2 > 0
H0: 1 - 2 = 0
Ha: 1 - 2 > 0
Ha: 1 - 2 < 0
Ha: 1 - 2  0
Test Statistic:
• Large-Sample Case
( x1  x2 )  ( 1  2 )
z
 12 n1   22 n2
•
Small-Sample Case
t
( x1  x2 )  ( 1  2 )
s2 (1 n1  1 n2 )
Slide 14
Example: Par, Inc.
Par, Inc. is a manufacturer of golf equipment. Par
has developed a new golf ball that has been designed to
provide “extra distance.” In a test of driving distance
using a mechanical driving device, a sample of Par golf
balls was compared with a sample of golf balls made by
Rap, Ltd., a competitor. The sample data is below.
Sample #1
Sample #2
Par, Inc.
Rap, Ltd.
Sample Size
n1 = 120 balls
n2 = 80 balls
x1 = 235 yards
x2 = 218 yards
Mean
Standard Deviation
s1 = 15 yards
s2 = 20 yards
Slide 15
Example: Par, Inc.
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Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
Can we conclude, using a .01 level of significance,
that the mean driving distance of Par, Inc. golf balls is
greater than the mean driving distance of Rap, Ltd.
golf balls?
1 = mean distance for the population of Par, Inc.
golf balls
2 = mean distance for the population of Rap, Ltd.
golf balls
Hypotheses:
H0: 1 - 2 < 0
Ha: 1 - 2 > 0
Slide 16
Example: Par, Inc.
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Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
Rejection Rule: Reject H0 if z > 2.33
z
( x1  x2 )  ( 1  2 )

2
1
n1


2
2
n2

(235 218)  0
17

 6.49
2
2
2.62
(15) (20)

120
80
Conclusion: Reject H0. We are at least 99% confident
that the mean driving distance of Par, Inc. golf balls is
greater than the mean driving distance of Rap, Ltd.
golf balls.
Slide 17