Business statistics, 7e, by groebner ch11

Business Statistics:
A Decision-Making Approach
7th Edition

Chapter 11
Hypothesis Tests for One and Two
Population Variances

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Chap 11-1


Chapter Goals
After completing this chapter, you should be able to:



Formulate and complete hypothesis tests for a single
population variance



Find critical chi-square distribution values from the chisquare table



Formulate and complete hypothesis tests for the
difference between two population variances



Use the F table to find critical F values

Business Statistics: A Decision-

Chap 11-2


Hypothesis Tests for Variances
Hypothesis Tests
for Variances

Tests for a Single
Population Variance

Tests for Two
Population Variances

Chi-Square test statistic

F test statistic

Business Statistics: A Decision-

Chap 11-3


Single Population
Hypothesis Tests for Variances

Tests for a Single
Population Variance

*

Chi-Square test statistic

Business Statistics: A Decision-

H0: σ2 = σ02
HA: σ2 ≠ σ02

Two tailed test

H0: σ2 ≥ σ02
HA: σ2 < σ02

Lower tail test

H0: σ2 ≤ σ02
HA: σ2 > σ02

Upper tail test

Chap 11-4


Chi-Square Test Statistic
Hypothesis Tests for Variances
The chi-squared test statistic for

a Single Population Variance is:

Tests for a Single
Population Variance

Chi-Square test statistic

(n − 1)s
χ =
σ2
2

*

2

where
χ2 = standardized chi-square variable
n = sample size
s2 = sample variance
σ2 = hypothesized variance

Business Statistics: A Decision-

Chap 11-5


The Chi-square Distribution


The chi-square distribution is a family of distributions, depending on degrees of freedom:



d.f. = n - 1

0 4 8 12 16 20 24 28

d.f. = 1

χ2

0 4 8 12 16 20 24 28

d.f. = 5

Business Statistics: A Decision-

χ2

0 4 8 12 16 20 24 28

χ2

d.f. = 15
Chap 11-6


Finding the Critical Value


The critical value,
chi-square table

, is found from the

χ 2α
Upper tail test:

H0: σ2 ≤ σ02
HA: σ2 > σ02
α
χ2
Do not reject H0

χ

Business Statistics: A Decision-

2

Reject H0
α

Chap 11-7


Example


A commercial freezer must hold the selected
temperature with little variation. Specifications call
for a standard deviation of no more than 4 degrees
(or variance of 16 degrees2). A sample of 16
freezers is tested and
yields a sample variance
of s2 = 24. Test to see
whether the standard
deviation specification
is exceeded. Use
α = .05

Business Statistics: A Decision-

Chap 11-8


Finding the Critical Value


Use the chi-square table to find the critical value:

χ 2α = 24.9958 (α = .05 and 16 – 1 = 15 d.f.)
The test statistic is:
2
(n
−
1)s
(16 − 1)24
2

χ =
=
= 22.5
2
σ
16

Since 22.5 < 24.9958,
do not reject H0
There is not significant
evidence at the α = .05 level
that the standard deviation
specification is exceeded

α = .05

χ2
Do not reject H0

Business Statistics: A Decision-

χ 2α

Reject H0

= 24.9958

Chap 11-9

Lower Tail or Two Tailed
Chi-square Tests
Lower tail test:

Two tail test:

H0: σ2 ≥ σ02
HA: σ2 < σ02

H0: σ2 = σ02
HA: σ2 ≠ σ02

α

α/2
α/2

χ2
Reject

χ

Do not reject H0
2

χ2
Reject

1-α

Business Statistics: A Decision-

Do not
reject H0

χ 21-α/2
(χ 2L)

Reject

χ 2α/2
(χ 2U)
Chap 11-10


Confidence Interval Estimate
for σ2


The confidence interval estimate for σ2 is

2
(n − 1)s2
(n
−
1)s
2
≤
σ
≤

2
χU
χL2

α/2
α/2

χ 21-α/2
(χ 2L)

χ 2α/2
(χ 2U)

χ2

Business Statistics: A Decision-

Where χ2L and χ2U are from the
χ2 distribution with n -1 degrees
of freedom

Chap 11-11


Example


A sample of 16 freezers yields a sample variance of s 2 = 24.



Form a 95% confidence interval for the population variance.

Business Statistics: A Decision-

Chap 11-12


Example
(continued)


Use the chi-square table to find χ2L and χ2U :
(α = .05 and 16 – 1 = 15 d.f.)

α/2=.025
α/2=.025
2

χ .975
(χ 2L)

χ .025
(χ 2U)

6.2621

27.4884

2

2
(n − 1)s 2
(n
−
1)s
2
≤
σ
≤
2
χU
χL2

(16 − 1)24
(16 − 1)24
2
≤σ ≤
27.4884
6.2621
13.096 ≤ σ 2 ≤ 57.489

We are 95% confident that the population variance is between 13.096
and 57.489 degrees2. (Taking the square root, we are 95% confident that
the population standard deviation is between 3.619 and 7.582 degrees.)

Business Statistics: A Decision-

Chap 11-13

F Test for Difference in Two
Population Variances
Hypothesis Tests for Variances

H0: σ12 = σ22
HA: σ12 ≠ σ22

Two tailed test

H0: σ12 ≥ σ22
HA: σ12 < σ22

Lower tail test

H0: σ12 ≤ σ22
HA: σ12 > σ22

*

Tests for Two
Population Variances

F test statistic
Upper tail test

Business Statistics: A Decision-

Chap 11-14

F Test for Difference in Two
Population Variances
Hypothesis Tests for Variances
The F test statistic is:
2
1
2
2

s
F=
s
s12

Where F has D1
numerator and D2
denominator
degrees of freedom

= Variance of Sample 1

Tests for Two
Population Variances

*

F test statistic

D1 = n1 - 1 = numerator degrees of freedom

s22

= Variance of Sample 2
D2 = n2 - 1 = denominator degrees of freedom

Business Statistics: A Decision-

Chap 11-15


The F Distribution


The F critical value is found from the F table



The are two appropriate degrees of freedom:
D1 (numerator) and D2 (denominator)

s12
F= 2
s2


where

D1 = n1 – 1 ; D2 = n2 – 1

In the F table,


numerator degrees of freedom determine the row



denominator degrees of freedom determine the column

Business Statistics: A Decision-

Chap 11-16


Formulating the F Ratio
s12
F= 2
s2

where

D1 = n1 – 1 ; D2 = n2 – 1



For a two-tailed test, always place the larger sample variance in the numerator



For a one-tailed test, consider the alternative hypothesis: place in the numerator the sample variance for the

population that is predicted (based on HA) to have the larger variance

Business Statistics: A Decision-

Chap 11-17


Finding the Critical Value
H0: σ12 ≥ σ22
HA: σ12 < σ22

H0: σ12 = σ22
HA: σ12 ≠ σ22

H0 : σ12 ≤ σ22
HA : σ12 > σ22
α

0

Do not
reject H0

Fα

Reject H0

α/2

F

rejection region
for a one-tail test is


s12
F = 2 > Fα
s2

0

Do not
reject H0

Fα/2

Reject H0

F

rejection region for
a two-tailed test is


s12
F = 2 > Fα / 2
s2

(where the larger sample variance in the numerator)

Business Statistics: A Decision-

Chap 11-18


F Test: An Example
You are a financial analyst for a brokerage firm. You want
to compare dividend yields between stocks listed on the
NYSE & NASDAQ. You collect the following data:
NYSE
NASDAQ
Number
2125
Mean
3.272.53
Std dev
1.301.16
Is there a difference in the
variances between the
NYSE
& NASDAQ at the α = 0.05 level?

Business Statistics: A Decision-

Chap 11-19


F Test: Example Solution




Form the hypothesis test:
H0: σ21 = σ22 (there is no difference between variances)
HA: σ21 ≠ σ22 (there is a difference between variances)
Find the F critical value for α = .05:
 Numerator:

D1 = n1 – 1 = 21 – 1 = 20
Denominator:
 D = n – 1 = 25 – 1 = 24
2
2




F.05/2, 20, 24 = 2.327

Business Statistics: A Decision-

Chap 11-20


F Test: Example Solution
(continued)


The test statistic is:

H0: σ12 = σ22
HA: σ12 ≠ σ22

s12 1.30 2
F= 2 =
= 1.256
2
s2 1.16

α/2 = .025

0



F = 1.256 is not greater than
the critical F value of 2.327, so
we do not reject H0



Conclusion: There is no evidence of a
difference in variances at α = .05

Business Statistics: A Decision-

Do not
reject H0

Reject H0

Fα/2
=2.327

Chap 11-21


Using EXCEL and PHStat
EXCEL


F test for two variances:


Data | Data Analysis | F-test: Two Sample for Variances

PHStat


Chi-square test for the variance:




PHStat | One-sample Tests | Chi-square Test for the Variance

F test for two variances:


PHStat | Two-sample Tests | F Test for Differences in Two

Variances

Business Statistics: A Decision-

Chap 11-22


Chapter Summary


Performed chi-square tests for the variance



Used the chi-square table to find chi-square critical values



Performed F tests for the difference between two population variances



Used the F table to find F critical values

Business Statistics: A Decision-

Chap 11-23