A Decision-Making Approach

7th Edition

Chapter 11

Hypothesis Tests and Estimation

for 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

chi-square 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-Making Approach, 7e © 2008 Prentice-Hall, Inc.

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-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Chap 11-3

Single Population

Hypothesis Tests for Variances

Tests for a Single

Population Variance

*

Chi-Square test statistic

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

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

2

σ

2

*

2

where

2 = standardized chi-square variable

n = sample size

s2 = sample variance

σ2 = hypothesized variance

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

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

2 0 4 8 12 16 20 24 28 2 0 4 8 12 16 20 24 28 2

d.f. = 5

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

d.f. = 15

Chap 11-6

Finding the Critical Value

2 , is found from the

The critical value,

chi-square table

Upper tail test:

Ho: σ2 16

H0: σ2 ≤ σ02

HA: σ2 > σ02

2

Do not reject H0

2

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

Reject H0

Chap 11-7

Example

Ho: σ2 16

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

= .05

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

Chap 11-8

Finding the Critical Value

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

2 = 24.9958 ( = .05 and 16 – 1 = 15 d.f.)

The test statistic is:

(n 1)s2 (16 1)24

2

22.5

2

σ

16

Since 22.5 < 24.9958,

do not reject H0

There is not significant

evidence at the = .05 level

that the standard deviation

specification is exceeded

= .05

2

Do not reject H0

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

2 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

/2

/2

2

Reject

21-

Do not reject H0

2

Reject

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

Do not

reject H0

21-/2

(2L)

Reject

2/2

(2U)

Chap 11-10

Confidence Interval Estimate

for σ2

The confidence interval estimate for σ2 is

(n 1)s2

(n 1)s 2

σ2

2

2

χU

χL...