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Inference Procedures forTwo Populations
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Independent versusDependent Samples
• Independent SamplesThe 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
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
(1 - ) * 100%Confidence Interval for
1 - 2
( X 1 X 2 ) Z / 2
12
n1
22
n2to ( X 1 X 2 ) Z / 2
12
n1
22
n2
For large samples where ’s are known
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
(1 - ) * 100%Confidence Interval for
1 - 2
( X 1 X 2 ) Z / 2
s12
n1
s22
n2to ( X 1 X 2 ) Z /2
s12
n1
s22
n2
For large samples where ’s are unknown
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Sample Sizes
n1 Z /2
2 s1 (s1 s2 )
E2
n2 Z /2
2 s2 ( s1 s2 )
E 2
nZ / 2
2 (s12 s2
2 )
E2
To minimizetotal samplesize:
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Hypothesis Testing for 1 and 2 (Large Samples)
ZX 1 X 21
2
n1
2
2
n2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Example 9.2
ZX 1 X 21
2
n1
2
2
n2
Define the HypothesisHo 1 2 (Texgas is less expensive)Ha 1 > 2 (Quik-Chek is less expensive)
Define the test statistic
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Define the rejection region ( = .05)Reject HO if Z > 1.645
Figure 9.4Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Evaluate the test statistic
Z X 1 X 21
2
n1
22
n2
X 1 X 2s1
2
n1
s22
n2
1.48 1.39
(.12)2
35 (.10)2
40
.09
.02573.50 Z*
State the conclusion: Quik-Chek stores do charge less for gasoline(on the average) than do Texgas stations.
Because 3.5 > 1.645 we reject HO
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Large Sample Tests for 1 and 2
Two-Tail TestHo 1 = 2
Ha 1 2
Reject HO if |Z| >Z /2
ZX 1 X 21
2
n1
2
2
n2Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Large Sample Tests for 1 and 2
One-Tail Test
Ho 1 2
Ha 1 > 2
Reject Ho if Z > Z
Ho 1 2
Ha 1 < 2
Reject Ho if Z < - Z
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
(1 - ) * 100% Confidence Interval for 1 - 2
(Small Independent Samples)
( X 1 X 2 ) t / 2,df
s12
n1
s22
n2to ( X 1 X 2 ) t / 2,df
s12
n1
s22
n2
df for t
s12
n1
s2
2
n2
2
s12
n1
2
n1 1
s22
n2
2
n2 1Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Hypothesis Testing for 1 - 2 (Small, Independent Samples)
Example 9.5
1. Ho 1 = 2
Ha 1 2
2. t X 1 X 2s1
2
n1
s2
2
n2
df for t
s12
n1
s2
2
n2
2
s12
n1
2
n1 1
s22
n2
2
n2 1
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
3. Reject HO if |t’| > t /2, df = t .05,22 = 1.717
t * 3.33 3.98
(.68)2
15
(.38)2
15
.65
.20 3.25
4.
Because |t’| = 3.25 > 1.717, we reject HO
5. There is a significant difference in the average blowout times for the two brandsIntroduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Comparing VariancesAssumptions
• Both populations are normal
• The samples are independent
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Comparing Variances
Figure 9.14
F s1
2
s22
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Hypothesis Test for 1 and 2
Two-Tail TestHo: 1 = 2
Ha: 1 2
F s1
2
s22
Reject Ho if F > F/2 ,v1 ,v2
or if F < F1-/2, v1 ,v2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Hypothesis Test for 1 and 2
One-Tail Test
Ho: 1 2
Ha: 1 > 2
Ho: 1 2
Ha: 1 < 2
F s1
2
s22
F s1
2
s22
Reject Ho if F > F ,v1 ,v2Reject Ho if F > F1- ,v1 ,v2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Confidence Interval for 12 /2
2
FR F.025,v1 ,v2and FL
1
F.025,v2 ,v1
s12 /s2
2
FRto
s12 /s2
2
FL
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing
Confidence Interval for d
d t / 2,n 1sd
nto d t / 2,n 1
sd
n
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College Publishing