Transcript Inference Procedures for Two Populations

Inference Procedures for
Two Populations
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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Independent versus
Dependent Samples
• Independent Samples
The occurrence of an observation in the first
sample has no effect on the value(s) in the other
sample.
• Dependent Samples or paired samples
– The occurrence of an observation in the first
sample has an impact on the corresponding
value in the second sample.
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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(1 - ) * 100%
Confidence Interval for
1 - 2
For large samples where ’s are known
( X1  X2 )  Z/ 2
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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12
n1

 22
n2
to
( X1  X2 )  Z/ 2
12
n1

 22
n2
(1 - ) * 100%
Confidence Interval for
1 - 2
For large samples where ’s are unknown
( X1  X2 )  Z/ 2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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s12
n1

s22
n2
to
( X1  X 2 )  Z /2
s12
n1

s22
n2
Sample Sizes
2
2
2
Z/
(s

s
2
1
2)
n
E2
To minimize
total sample
size:
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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n1 
n2 
Z2 /2 s1 (s1  s2 )
E2
2
Z /2
s2 ( s1  s2 )
E2
Hypothesis Testing for 1 and 2
(Large Samples)
Z
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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X1  X 2
 12  22

n1 n2
Example 9.2
Define the Hypothesis
Ho 1  2
Ha 1 > 2
(Texgas is less expensive)
(Quik-Chek is less expensive)
Define the test statistic
Z
Introduction to Business Statistics, 5e
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X1  X 2
 12  22

n1 n2
Define the rejection region ( = .05)
Reject HO if Z > 1.645
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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Figure 9.4
Evaluate the test statistic
Z

X1  X2
 12  22

n1 n2

X1  X2
s12 s22

n1 n 2

1.481.39
(.12) 2 (.10) 2

35
40
.09
 3.50  Z*
.0257
Because 3.5 > 1.645 we reject HO
State the conclusion: Quik-Chek stores do charge
less for gasoline(on the average) than do
Texgas stations.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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Large Sample Tests for 1 and 2
Two-Tail Test
Ho 1 = 2
Ha 1  2
Reject HO if |Z| > Z /2
Z
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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X1  X 2
 12  22

n1 n2
Large Sample Tests for 1 and 2
One-Tail Test
Ho 1  2
Ha 1 > 2
Reject Ho if Z > Z
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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Ho 1  2
Ha 1 < 2
Reject Ho if Z < - Z
(1 - ) * 100% Confidence
Interval for 1 - 2
(Small Independent Samples)
( X1  X2 )  t/ 2,df
s12
n1

s22
n2
to
( X1  X2 )  t/ 2,df
s12 s22 
 


n1 n2 

df for t  
2
2
 s12 
 s22 







n
n
 1 
 2 


n1 1
n2  1
2
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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s12
n1

s22
n2
Hypothesis Testing for 1 - 2
(Small, Independent Samples)
Example 9.5
1.
Ho 1 = 2
Ha 1  2
s12 s22 
 


n
n
 1
2 

df for t  
2
2
 s12 
 s22 







n
n
 1 
 2 


n1 1
n2  1
2
2.
Introduction to Business Statistics, 5e
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t  
X1  X2
s12 s22

n1 n2
3.
Reject HO if |t’| > t /2, df = t .05,22 = 1.717
4.
3.33 3.98
.65

 3.25
.20
(.6 8) 2 (.38) 2

15
15
t * 
Because |t’| = 3.25 > 1.717, we reject HO
5.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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There is a significant difference in the
average blowout times for the two brands
Comparing Variances
Assumptions
• Both populations are normal
• The samples are independent
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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Comparing Variances
s12
F 2
s2
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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Figure 9.14
Hypothesis Test for 1 and 2
Two-Tail Test
Ho: 1 = 2
Ha: 1  2
s12
F 2
s2
F/2 ,v1 ,v 2
or if F < F1-/2, v1 ,v 2
Reject Ho if F >
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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Hypothesis Test for 1 and 2
One-Tail Test
Ho: 1  2
Ha: 1 > 2
Ho: 1  2
Ha: 1 < 2
s12
F 2
s2
s12
F 2
s2
Reject Ho if F >
Introduction to Business Statistics, 5e
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F ,v1 ,v 2
Reject Ho if F >
F1- ,v1 ,v 2
Confidence Interval for 12 /22
FR  F.025,v1 ,v2
s12 /s22
FR
Introduction to Business Statistics, 5e
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and
to
FL 
1
F.025,v2 ,v1
s12 /s22
FL
Confidence Interval for d
d t/ 2,n1
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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sd
n
to d  t/ 2,n1
sd
n
Hypothesis Test for d
Ho: d = 0
Ha: d  0
tD 
Introduction to Business Statistics, 5e
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d  D0
sd / n