INFERENCE: TWO POPULATIONS

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Transcript INFERENCE: TWO POPULATIONS

統計學
Spring 2004
授課教師:統計系余清祥
日期:2004年3月2日
第三週:比較期望值
Slide 1
Chapter 10
Comparisons Involving Means




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
Inferences about the Difference between the Means
of Two Populations: Matched Samples
Inferences about the Difference between the
Proportions of Two Populations:
Slide 2
Estimation of the Difference Between the Means
of Two Populations: Independent Samples




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 3
Point Estimator of the Difference Between
the Means of Two Populations





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.
x2
Let x1 equal the mean of sample 1 and
equal the
mean of sample 2.
The point estimator of the difference between the
x1 1 and
x2 2 is
means of the populations
.
Slide 4
Sampling Distribution ofx1  x2

Properties of the Sampling Distribution xof
1  x2
• Expected Value
E ( x1  x2 )  1   2
• Standard Deviation
 x1  x2 
12
n1

 22
n2
where: 1 = standard deviation of population 1
2 = standard deviation of population 2
n1 = sample size from population 1
n2 = sample size from population 2
Slide 5
Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and n2 > 30)

Interval Estimate with 1 and 2 Known
where:

x1  x2  z / 2  x1  x2
1 -  is the confidence coefficient
Interval Estimate with 1 and 2 Unknown
x1  x2  z / 2 sx1  x2
where:
sx1  x2
s12 s22


n1 n2
Slide 6
Example: Par, Inc.

Interval Estimate of 1 - 2: Large-Sample Case
Par, Inc. is a manufacturer of golf equipment and
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 statistics appear on the next slide.
Slide 7
Example: Par, Inc.

Interval Estimate of 1 - 2: Large-Sample Case
• Sample Statistics
Sample Size
Mean
Standard Dev.
Sample #1
Par, Inc.
n1 = 120 balls
x1 = 235 yards
s1 = 15 yards
Sample #2
Rap, Ltd.
n2 = 80 balls
x2 = 218 yards
s2 = 20 yards
Slide 8
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 9
Point Estimator of the Difference
Between the Means of Two Populations
Population 1
Par, Inc. Golf Balls
Population 2
Rap, Ltd. Golf Balls
1 = mean driving
2 = mean driving
distance of Par
golf balls
distance of Rap
golf balls
m1 – 2 = difference between
the mean distances
Simple random sample
of n1 Par golf balls
Simple random sample
of n2 Rap golf balls
x1 = sample mean distance
for sample of Par golf ball
x2 = sample mean distance
for sample of Rap golf ball
x1 - x2 = Point Estimate of m1 – 2
Slide 10
Example: Par, Inc.

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 11
Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30 and/or n2 < 30)

Interval Estimate with  2 Known
x1  x2  z / 2  x1  x2
where:
 x1  x2
1 1
  (  )
n1 n2
2
Slide 12
Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30 and/or n2 < 30)

Interval Estimate with  2 Unknown
x1  x2  t / 2 sx1  x2
where:
sx1  x2
1 1
 s (  )
n1 n2
2
2
2
(
n

1
)
s

(
n

1
)
s
1
2
2
s2  1
n1  n2  2
Slide 13
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 statistics are:
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 14
Example: Specific Motors

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 15
Example: Specific Motors

95% Confidence Interval Estimate of the Difference
Between Two Population Means: Small-Sample Case
We will make the following assumptions:
• The miles per gallon rating must be normally
distributed for both the M car and the J car.
• 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 16
Example: Specific Motors

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
1 1
1 1
s (  )  2. 5  2.101 5. 28(  )
n1 n2
12 8
2
= 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 17
Hypothesis Tests About the Difference
Between the Means of Two Populations:
Independent Samples

Hypotheses
H0: 1 - 2 < 0
Ha: 1 - 2 > 0

H0: 1 - 2 > 0
Ha: 1 - 2 < 0
Test Statistic
Large-Sample
z
( x1  x2 )  ( 1   2 )
12 n1   22 n2
H0: 1 - 2 = 0
Ha: 1 - 2  0
Small-Sample
t
( x1  x2 )  ( 1   2 )
s2 (1 n1  1 n2 )
Slide 18
Example: Par, Inc.

Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
Par, Inc. is a manufacturer of golf equipment and 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
statistics appear on the next slide.
Slide 19
Example: Par, Inc.

Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
• Sample Statistics
Sample Size
Mean
Standard Dev.
Sample #1
Par, Inc.
n1 = 120 balls
x1 = 235 yards
s1 = 15 yards
Sample #2
Rap, Ltd.
n2 = 80 balls
x2 = 218 yards
s2 = 20 yards
Slide 20
Example: Par, Inc.

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 21
Example: Par, Inc.

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 )
12
n1

 22
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 22
Example: Specific Motors

Hypothesis Tests About the Difference Between the
Means of Two Populations: Small-Sample Case
Can we conclude, using a .05 level of
significance, that the miles-per-gallon (mpg)
performance of M cars is greater than the miles-pergallon performance of J cars?
1 = mean mpg for the population of M cars
2 = mean mpg for the population of J cars
• Hypotheses H0: 1 - 2 < 0
Ha: 1 - 2 > 0
Slide 23
Example: Specific Motors

Hypothesis Tests About the Difference Between the
Means of Two Populations: Small-Sample Case
• Rejection Rule
Reject H0 if t > 1.734
(a = .05, d.f. = 18)
• Test Statistic
t
where:
( x1  x2 )  ( 1   2 )
s2 (1 n1  1 n2 )
(n1  1)s12  (n2  1)s22
s 
n1  n2  2
2
Slide 24
Inference About the Difference Between the
Means of Two Populations: Matched Samples



With a matched-sample design each sampled item
provides a pair of data values.
The matched-sample design can be referred to as
blocking.
This design often leads to a smaller sampling error
than the independent-sample design because
variation between sampled items is eliminated as a
source of sampling error.
Slide 25
Example: Express Deliveries

Inference About the Difference Between the Means of
Two Populations: Matched Samples
A Chicago-based firm has documents that must
be quickly distributed to district offices throughout
the U.S. The firm must decide between two delivery
services, UPX (United Parcel Express) and INTEX
(International Express), to transport its documents.
In testing the delivery times of the two services, the
firm sent two reports to a random sample of ten
district offices with one report carried by UPX and
the other report carried by INTEX.
Do the data that follow indicate a difference in
mean delivery times for the two services?
Slide 26
Example: Express Deliveries
District Office
Seattle
Los Angeles
Boston
Cleveland
New York
Houston
Atlanta
St. Louis
Milwaukee
Denver
Delivery Time (Hours)
UPX
INTEX
Difference
32
30
19
16
15
18
14
10
7
16
25
24
15
15
13
15
15
8
9
11
7
6
4
1
2
3
-1
2
-2
5
Slide 27
Example: Express Deliveries

Inference About the Difference Between the Means of
Two Populations: Matched Samples
Let d = the mean of the difference values for the
two delivery services for the population of
district offices
• Hypotheses
• Rejection Rule
H0: d = 0, Ha: d 
Assuming the population of difference values is
approximately normally distributed, the t
distribution with n - 1 degrees of freedom applies.
With  = .05, t.025 = 2.262 (9 degrees of freedom).
Reject H0 if t < -2.262 or if t > 2.262
Slide 28
Example: Express Deliveries

Inference About the Difference Between the Means of
Two Populations: Matched Samples
 di ( 7  6... 5)
d 

 2. 7
n
10
2
76.1
 ( di  d )
sd 

 2. 9
n 1
9
d  d
2. 7  0
t

 2. 94
sd n 2. 9 10
• Conclusion
Reject H0.
There is a significant difference between the mean
delivery times for the two services.
Slide 29
Inferences About the Difference
Between the Proportions of Two Populations



Sampling Distribution of p1  p2
Interval Estimation of p1 - p2
Hypothesis Tests about p1 - p2
Slide 30
Sampling Distribution of p1  p2

Expected Value
E ( p1  p2 )  p1  p2

Standard Deviation
 p1  p2 

p1 (1  p1 ) p2 (1  p2 )

n1
n2
Distribution Form
If the sample sizes are large (n1p1, n1(1 - p1), n2p2,
and n2(1 - p2) are all greater than or equal to 5), the
sampling distribution of
p1  p2 can be approximated
by a normal probability distribution.
Slide 31
Interval Estimation of p1 - p2

Interval Estimate
p1  p2  z / 2  p1  p2

Point Estimator of  p1  p2
s p1  p2 
p1 (1  p1 ) p2 (1  p2 )

n1
n2
Slide 32
Example: MRA
MRA (Market Research Associates) is conducting
research to evaluate the effectiveness of a client’s new
advertising campaign. Before the new campaign
began, a telephone survey of 150 households in the
test market area showed 60 households “aware” of
the client’s product. The new campaign has been
initiated with TV and newspaper advertisements
running for three weeks. A survey conducted
immediately after the new campaign showed 120 of
250 households “aware” of the client’s product.
Does the data support the position that the
advertising campaign has provided an increased
awareness of the client’s product?
Slide 33
Example: MRA

Point Estimator of the Difference Between the
Proportions of Two Populations
120 60
p1  p2  p1  p2 

. 48. 40 . 08
250 150
p1 = proportion of the population of households
“aware” of the product after the new campaign
p2 = proportion of the population of households
“aware” of the product before the new
p1
campaign
= sample proportion of households “aware” of the
p2 product after the new campaign
= sample proportion of households “aware” of the
product before the new campaign
Slide 34
Example: MRA

Interval Estimate of p1 - p2: Large-Sample Case
For = .05, z.025 = 1.96:
. 48(.52) . 40(. 60)
. 48. 40  1. 96

250
150
.08 + 1.96(.0510)
.08 + .10
or -.02 to +.18
• Conclusion
At a 95% confidence level, the interval estimate of
the difference between the proportion of households
aware of the client’s product before and after the new
advertising campaign is -.02 to +.18.
Slide 35
Hypothesis Tests about p1 - p2

Hypotheses
H0: p1 - p2 < 0
Ha: p1 - p2 > 0

Test statistic
z

( p1  p2 )  ( p1  p2 )
 p1  p2
Point Estimator of  p1  p2 where p1 = p2
s p1  p2  p (1  p )(1 n1  1 n2 )
where:
n1 p1  n2 p2
p
n1  n2
Slide 36
Example: MRA

Hypothesis Tests about p1 - p2
Can we conclude, using a .05 level of
significance, that the proportion of households aware
of the client’s product increased after the new
advertising campaign?
p1 = proportion of the population of households
“aware” of the product after the new campaign
p2 = proportion of the population of households
“aware” of the product before the new
campaign
• Hypotheses
H0: p1 - p2 < 0
Ha: p1 - p2 > 0
Slide 37
Example: MRA

Hypothesis Tests about p1 - p2
• Rejection Rule
Reject H0 if z > 1.645
• Test Statistic
250(. 48)  150(. 40) 180
p

. 45
250  150
400
s p1  p2  . 45(.55)( 1
 1 ) . 0514
250 150
(. 48. 40)  0
. 08
z

 1.56
. 0514
. 0514
• Conclusion
Do not reject H0.
Slide 38
End of Chapter 10
Slide 39