2/7/2011
1
TwoSample Tests
Chapter 11
Page 452
Objectives
1. Comparing The Means of Two Independent
Populations
•Ztest for the difference between two means.
•PooledVariance ttest for the difference between
Two Means.
2. Comparing The Means of Two Related Populations
3. Comparing Two Population Proportions
•ZTest
4. FTest For The Difference Between Two Variances
Hypothesis Test
One Population
Hypothesis Test
Two Populations
Sample
Independent Sample
dependent
2 MEANS2 Proportions
Check the 4
conditions
If yes
Normal?
No
STOP
Equal V2?unequal V2
V1and V2known?
Ztest Pooled Variance ttest
Ztest
Paired ttest
No
YesYe s
No
Ttest
“not” Pooled variance
Comparing Dependent and Independent
Samples
Independent SamplesIndependent Samples
•The samples chosen at random are
not relatednot relatedto each other.
•We wish to study the mean incomes
Dependent SampleDependent Sample
•Dependent samples are
characterized by a measurement,
then some type of intervention,
followed b
another measurement.
oGGGGGGGkG
GGf
.
•We select a random sample of 28
employees from the Company X and
a sample of 19 employees in
Company Y.
•A person cannot be an employee in
both companies.
•The samples are independentindependent,
that is, unrelated.
•Paired samples are also
dependent because the same
individual or item is a member of
both samples.
•Examples: 10 participants in a
marathon were weighed prior to
and after competing in the race.
•We wish to study the mean
amount of weight loss.
There are two types of dependent samples:
•Characterized by a Measurement,
followed by an intervention of
some type, and then another
measurement
•Called a “before” and “after” study.
•Example:
Suppose we want to show that, by
placing speakers in the production
area and playing soothing music,
•Characterized by matching or
pairing of the observations.
•Example:
Suppose you are selling a house
and you asked for two
appraisals of your property.
Suppose an industrial
psychologist wishes to study
the intellectual similarities of
production.
We begin by selecting a sample of
workers and measuring their
output under current conditions,
The speakers are then installed in
the production area, and we again
measure the output of the same
workers.
There are 2 measurements: (1)
before placing the speakers
(2) After placing the speakers.
newly married couples. She
selects a sample of
newlyweds.
Next she administers a standard
intelligence test to both the
man and woman to determine
the difference in the scores
Notice the matching that
occurred: comparing the
scores of the man and the
woman.
Comparing Dependent and Independent
Samples
2
2
2
1
2
1
2121
calc
n
1
n
1
)()xx(
z
Independent SamplesIndependent Samples Dependent SampleDependent Sample
ztest: Test of two indep
pop, VKnown
n
S
D
t
D
D
calc
Paired ttest:
Test of two dep pop, Vequal but
unknown
Where D is “difference scores”
=

Section 11.1Section 11.2
)1n()1n(
s)1n(s)1n(
s
21
2
22
2
11
2
p
)
n
1
n
1
(s
)()xx(
t
21
2
p
2121
calc
ttest: Test of two indep pop,
Vequal but unknown
df=(n11)+(n21)=n1+n22
ttest: Test of two indep pop,
Vunequal and unknown
2
2
2
1
2
1
2121
calc
n
s
n
s
)()xx(
t
)1n(n
s
)1n(n
s
)
n
s
()
n
s
(
df
2
2
2
4
2
1
2
1
4
1
2
2
2
1
2
1
The sevenstep Method of Hypothesis Testing
Steps:
1. Define the parameters: P1and P2
2. State the hypothesis
Ho: P1= P2 (or Ho: P1P2 = 0)
Ha: P1 P2 (or Ha: P1P2 0)
3. Determine the appropriate test (refer to the flowchart)
4. State the conditions/assumptions: (1) Normal? (2) Equal variance?
Determine the critical value that divide the rejection and non
.

6. Calculate the test statistics value
7. Make the statistical decision and Managerial conclusion
To reject Ho or DO NOT reject Ho
Compare the test statistics with the critical value (critical value approach)
or
compare pvalue with .(pvalue approach)
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