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

Hypothesis Test for d

Ho: d = 0Ha: d 0

tD d D0

sd / nIntroduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing