### Chapter 8 Inferences from Two Samples

```ELEMENTARY
Chapter 8
MARIO F. TRIOLA
STATISTICS
Inferences from Two Samples
EIGHTH
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
EDITION 1
Chapter 8
Inferences from Two Samples
8-1 Overview
8-2 Inferences about Two Means: Independent
and Large Samples
8-3 Inferences about Two Means: Matched Pairs
8-5 Comparing Variation in Two Samples
Independent
and Small Samples
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
8-1
Overview
There are many important and meaningful
situations in which it becomes necessary
to compare two sets of sample data.
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
8-2
Independent and
Large Samples
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
Definitions
Two Samples: Independent
The sample values selected from one
population are not related or somehow paired
with the sample values selected from the
other population.
If the values in one sample are related to the
values in the other sample, the samples are
dependent. Such samples are often referred
to as matched pairs or paired samples.
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Assumptions
1. The two samples are independent.
2. The two sample sizes are large. That
is, n1 > 30 and n2 > 30.
3. Both samples are simple
random samples.
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Hypothesis Tests
Test Statistic for Two Means:
Independent and Large Samples
z
=
(x1 - x2) - (µ1 - µ2)

2.
1
n1

2
2
+ n
2
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Hypothesis Tests
Test Statistic for Two Means:
Independent and Large Samples
 and 
If  and  are not known, use s1 and s2
in their places. provided that both
samples are large.
P-value:
Use the computed value of the test
statistic z, and find the P-value by following
the same procedure summarized in Figure 7-8.
Critical values:
Based on the significance level ,
find critical values by using the
procedures introduced in Section 7-2.
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Coke Versus Pepsi
Data Set 1 in Appendix B includes the weights (in
pounds) of samples of regular Coke and regular
Pepsi. Sample statistics are shown. Use the 0.01
significance level to test the claim that the mean
weight of regular Coke is different from the mean
weight of regular Pepsi.
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Coke Versus Pepsi
Data Set 1 in Appendix B includes the weights (in
pounds) of samples of regular Coke and regular
Pepsi. Sample statistics are shown. Use the 0.01
significance level to test the claim that the mean
weight of regular Coke is different from the mean
weight of regular Pepsi.
Regular Coke
Regular Pepsi
n
36
36
x
0.81682
0.82410
s
0.007507
0.005701
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Coke Versus Pepsi
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Coke Versus Pepsi
Claim: 1  2
Ho :  1 =  2
H1 :  1   2
 = 0.01
Reject H0
Z = - 2.575
Fail to reject H0
1 -  = 0
Reject H0
Z = 2.575
or Z = 0
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Coke Versus Pepsi
Test Statistic for Two Means:
Independent and Large Samples
z
=
(x1 - x2) - (µ1 - µ2)

2.
1
n1

2
2
+ n
2
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Coke Versus Pepsi
Test Statistic for Two Means:
Independent and Large Samples
z
=
(0.81682 - 0.82410) - 0
0.0075707 2
36
+
0.005701 2
36
= - 4.63
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Coke Versus Pepsi
Claim: 1  2
Ho :  1 =  2
H1 :  1   2
 = 0.01
Reject H0
sample data:
z = - 4.63
Z = - 2.575
Fail to reject H0
1 -  = 0
Reject H0
Z = 2.575
or Z = 0
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
Coke Versus Pepsi
Claim: 1  2
Ho :  1 =  2
H1 :  1   2
There is significant evidence to support
the claim that there is a difference
between the mean weight of Coke and
the mean weight of Pepsi.
 = 0.01
Reject H0
Fail to reject H0
Reject H0
Reject Null
sample data:
z = - 4.63
Z = - 2.575
1 -  = 0
Z = 2.575
or Z = 0
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
Confidence Intervals
(x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
Confidence Intervals
(x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E
where E =
z

2
1
n1

2
2
+ n
2
Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18
```