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Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin
Statistical Inferences Based on
Two Samples
Chapter 10
10-2
Chapter Outline
10.1 Comparing Two Population Means by Using Independent Samples: Variances Known
10.2 Comparing Two Population Means by Using Independent Samples: Variances Unknown
10.3 Paired Difference Experiments10.4 Comparing Two Population Proportions
by Using Large, Independent Samples10.5 Comparing Two Population Variances
by Using Independent Samples
10-3
10.1 Comparing Two Population Means by Using Independent Samples: Variances Known
Suppose a random sample has been taken from each of two different populations
Suppose that the populations are independent of each other Then the random samples are independent of
each other
Then the sampling distribution of the difference in sample means is normally distributed
10-4
Sampling Distribution of theDifference of Two Sample Means #1
Suppose population 1 has mean µ1 and variance σ1
2
From population 1, a random sample of size n1 is selected which has mean 1 and variance s1
2
Suppose population 2 has mean µ2 and variance σ2
2
From population 2, a random sample of size n2 is selected which has mean 2 and variance s2
2
Then the sample distribution of the difference of two sample means…
10-5
Sampling Distribution of theDifference of Two Sample Means #2
Is normal, if each of the sampled populations is normalApproximately normal if the sample
sizes n1 and n2 are large
Has mean µ1–2 = µ1 – µ2
Has standard deviation
2
22
1
21
21 nnxx
10-6
Sampling Distribution of theDifference of Two Sample Means #3
Figure 10.1
10-7
z-Based Confidence Interval for the Difference in Means (Variances Known)
A 100(1 – ) percent confidence interval for the difference in populations µ1–µ2 is
2
22
1
21
221 nnzxx
10-8
z-Based Test About the Difference in Means (Variances Known)
Test the null hypothesis aboutH0: µ1 – µ2 = D0
D0 = µ1 – µ2 is the claimed difference between the population means
D0 is a number whose value varies depending on the situation
Often D0 = 0, and the null means that there is no difference between the population means
10-9
z-Based Test About the Difference in Means (Variances Known)
Use the notation from the confidence interval statement on a prior slide
Assume that each sampled population is normal or that the samples sizes n1 and n2 are large
10-10
Test Statistic (Variances Known)
The test statistic is
The sampling distribution of this statistic is a standard normal distribution
If the populations are normal and the samples are independent ...
2
22
1
21
021
nn
Dxxz
10-11
z-Based Test About the Difference in Means (Variances Known)
Reject H0: µ1 – µ2 = D0 in favor of a particular alternative hypothesis at a level of significance if the appropriate rejection point rule holds or if the corresponding p-value is less than
Rules are on the next slide…
10-12
z-Based Test About the Difference in Means (Variances Known) Continued
10-13
Example 10.2: The Bank Customer Waiting Time Case
21.14
1009.1
1007.4
014.579.8
0:
0:
2
22
1
21
021
21
210
nn
Dxxz
H
H
a
10-14
10.2 Comparing Two Population Means by Using Independent Samples: Variances Unknown
Generally, the true values of the population variances σ1
2 and σ22
are not knownThey have to be estimated from
the sample variances s12 and s2
2, respectively
10-15
Comparing Two Population Means Continued
Also need to estimate the standard deviation of the sampling distribution of the difference between sample means
Two approaches:1. If it can be assumed that σ1
2 = σ22 = σ2,
then calculate the “pooled estimate” of σ2
2. If σ12 ≠ σ2
2, then use approximate methods
10-16
Pooled Estimate of σ2
21
2
21
222
2112
11
2
11
21 nns
nn
snsns
pxx
p
10-17
t-Based Confidence Interval for the Difference in Means (Variances Unknown)
2
11
11
21
222
2112
21
2221
nn
snsns
nnstxx
p
p
10-18
Example 10.3: The Catalyst Comparison Case
22.91,38.3042.302.750811
42.305
1
5
11.435
11
1.435255
2.4841538615
2
11
21
21
21
21
2
21
222
2112
xx
pxx
p
xx
nns
nn
snsns
10-19
t-Based Test About the Difference in Means: Variances Equal
21
2
021
11
nns
Dxx
p
10-20
Example 10.4: The Catalyst Comparison Case
6087.4
51
51
1.435
02.750811
11
0:
0:
21
2
021
211
210
nns
Dxxt
H
H
p
10-21
t-Based Confidence Intervals and Tests for Differences with Unequal Variances
11 2
2
222
1
2
121
2
2221
21
2
22
1
21
021
2
22
1
21
/221
n/ns
n/ns
/ns/nsdf
ns
ns
Dxxt
n
s
n
stxx
10-22
10.3 Paired Difference Experiments
Before, drew random samples from two different populations
Now, have two different processes (or methods)
Draw one random sample of units and use those units to obtain the results of each process
10-23
Paired Difference Experiments Continued
For instance, use the same individuals for the results from one process vs. the results from the other processE.g., use the same individuals to compare
“before” and “after” treatmentsUsing the same individuals, eliminates
any differences in the individuals themselves and just comparing the results from the two processes
10-24
Paired Difference Experiments #3
Let µd be the mean of population of paired differences µd = µ1 – µ2, where µ1 is the mean of population 1
and µ2 is the mean of population 2
Let d̄ and sd be the mean and standard deviation of a sample of paired differences that has been randomly selected from the population d̄ is the mean of the differences between pairs of
values from both samples
10-25
t-Based Confidence Interval for Paired Differences in Means
n/s
Ddt=
n
std
d
d/
0
2
10-26
Paired Differences Testing Rules
10-27
Example 10.6 and 10.7: The Repair Cost Comparison Case
2053.475033.
08.
0:
0:
3346.,2654.17
5033.447.28.
0
2
n/s
Ddt=
H
H
n
std
d
da
do
d/
10-28
10.4 Comparing Two Population Proportions by Using Large, Independent Samples
Select a random sample of size n1 from a population, and let p̂1 denote the proportion of units in this sample that fall into the category of interest
Select a random sample of size n2 from another population, and let p̂2 denote the proportion of units in this sample that fall into the same category of interest
Suppose that n1 and n2 are large enough n1·p1 ≥ 5, n1·(1 - p1) ≥ 5, n2·p2 ≥ 5, and n2·(1 –
p2) ≥ 5
10-29
Comparing Two Population Proportions Continued
Then the population of all possible values of p̂1 - p̂2
Has approximately a normal distribution if each of the sample sizes n1 and n2 is large
Has mean µp̂1 - p̂2 = p1 – p2
Has standard deviation 2
22
1
11 1121 n
pp
n
ppp̂p̂
10-30
Difference of Two Population Proportions
21 ˆˆ
021
2
22
1
11221
ˆˆ
ˆ1ˆˆ1ˆˆˆ
pp
Dppz=
n
pp
n
ppzpp
10-31
Example 10.9 and 10.10: The Advertising Media Case
2673.8
000,11
000,11
7145.17145.
0798.631.
11ˆ1ˆ
ˆˆˆˆ
1281.,2059.
000,1
202.798.
000,1
369.631.96.1798.631.
ˆ1ˆˆ1ˆˆˆ
21
021
ˆˆ
021
2
22
1
11221
21
z
nnpp
DppDppz=
n
pp
n
ppzpp
pp
10-32
10.5 Comparing Two Population Variances Using Independent Samples
Population 1 has variance σ12 and population 2
has variance σ22
The null hypothesis H0 is that the variances are the same H0: σ1
2 = σ22
The alternative is that one is smaller than the other That population has less variable measurements Suppose σ1
2 > σ22
More usual to normalize
Test H0: σ12/σ2
2 = 1 vs. σ12/σ2
2 > 1
10-33
Comparing Two Population Variances Using Independent Samples Continued
Reject H0 in favor of Ha if s12/s2
2 is significantly greater than 1
s12 is the variance of a random of size n1 from
a population with variance σ12
s22 is the variance of a random of size n2 from
a population with variance σ22
To decide how large s12/s2
2 must be to reject H0, describe the sampling distribution of s1
2/s22
The sampling distribution of s12/s2
2 is the F distribution
10-34
F Distribution
Figure 10.13
10-35
F Distribution
The F point F is the point on the horizontal axis under the curve of the F distribution that gives a right-hand tail area equal to The value of F depends on a (the size of the
right-hand tail area) and df1 and df2
Different F tables for different values of Tables A.5 for = 0.10 Tables A.6 for = 0.05 Tables A.7 for = 0.025 Tables A.8 for = 0.01
10-36
Example 10.11: The Catalyst Comparison Case
2544.1386
2.484
:
:
21
22
21
22
22
21
22
210
s
sF
H
H
a