Section 10.3 Comparing Two Variances Larson/Farber 4th ed1.

Section 10.3Section 10.3Comparing Two Variances

Larson/Farber 4th ed 1

Section 10.3 ObjectivesSection 10.3 ObjectivesInterpret the F-distribution and

use an F-table to find critical values

Perform a two-sample F-test to compare two variances


FF-Distribution-DistributionLet represent the sample

variances of two different populations.

If both populations are normal and the population variances are equal, then the sampling distribution of

is called an F-distribution. Larson/Farber 4th ed 3

2 21 2 and s s

2 21 2 and σ σ

2122

sF

s

Properties of the Properties of the FF--DistributionDistribution1. The F-distribution is a family of curves

each of which is determined by two types of degrees of freedom: ◦ The degrees of freedom corresponding to

the variance in the numerator, denoted d.f.N

◦ The degrees of freedom corresponding to the variance in the denominator, denoted d.f.D

2. F-distributions are positively skewed.3. The total area under each curve of an F-

distribution is equal to 1.Larson/Farber 4th ed 4

Properties of the Properties of the FF--DistributionDistribution

4. F-values are always greater than or equal to 0.5. For all F-distributions, the mean value of F is

approximately equal to 1.


d.f.N = 1 and d.f.D = 8

d.f.N = 8 and d.f.D = 26

d.f.N = 16 and d.f.D = 7

d.f.N = 3 and d.f.D = 11

F1 2 3 4

Critical Values for the Critical Values for the FF--DistributionDistribution1. Specify the level of significance .

2. Determine the degrees of freedom for the numerator, d.f.N.

3. Determine the degrees of freedom for the denominator, d.f.D.

4. Use Table 7 in Appendix B to find the critical value. If the hypothesis test is

a. one-tailed, use the F-table.

b. two-tailed, use the ½ F-table.


Example: Finding Critical Example: Finding Critical FF--ValuesValuesFind the critical F-value for a right-tailed test when α = 0.05, d.f.N = 6 and d.f.D = 29.


The critical value is F0 = 2.43.

Solution:

Example: Finding Critical Example: Finding Critical FF--ValuesValuesFind the critical F-value for a two-tailed test when α = 0.05, d.f.N = 4 and d.f.D = 8.


Solution:•When performing a two-tailed hypothesis test using the F-distribution, you need only to find the right-tailed critical value. •You must remember to use the ½α table.

1(0.05) 0.025

2

1

2

Solution: Finding Critical Solution: Finding Critical FF-Values-Values

½α = 0.025, d.f.N = 4 and d.f.D = 8


The critical value is F0 = 5.05.

Two-Sample Two-Sample FF-Test for -Test for VariancesVariancesTo use the two-sample F-test for comparing two population variances, the following must be true.1.The samples must be randomly selected.2.The samples must be independent.3.Each population must have a normal distribution.


Two-Sample Two-Sample FF-Test for -Test for VariancesVariancesTest Statistic


2122

sF

s

where represent the sample variances with

• The degrees of freedom for the numerator is d.f.N = n1 – 1 where n1 is the size of the sample having variance

• The degrees of freedom for the denominator is d.f.D = n2 – 1, and n2 is the size of the sample having variance

2 21 2 and s s

2 21 2.s s

21 .s

22.s

TwoTwo--Sample Sample FF--Test for Test for VariancesVariances


1. Identify the claim. State the null and alternative hypotheses.

2. Specify the level of significance.

3. Identify the degrees of freedom.

4. Determine the critical value.

State H0 and Ha.

Identify .

Use Table 7 in Appendix B.

d.f.N = n1 – 1 d.f.D = n2 – 1

In Words In Symbols

TwoTwo--Sample Sample FF--Test for Test for VariancesVariances

2122

sF

s


If F is in the rejection region, reject H0. Otherwise, fail to reject H0.

5. Determine the rejection region.

6. Calculate the test statistic.

7. Make a decision to reject or fail to reject the null hypothesis.

8. Interpret the decision in the context of the original claim.

In Words In Symbols

Example: Performing a Two-Example: Performing a Two-Sample Sample FF-Test-Test

A restaurant manager is designing a system that is intended to decrease the variance of the time customers wait before their meals are served. Under the old system, a random sample of 10 customers had a variance of 400. Under the new system, a random sample of 21 customers had a variance of 256. At α = 0.10, is there enough evidence to convince the manager to switch to the new system? Assume both populations are normally distributed.


Solution: Performing a Two-Solution: Performing a Two-Sample Sample FF-Test-Test


• H0:

• Ha:

• α =

• d.f.N= d.f.D=

• Rejection Region:

• Test Statistic:

• Decision:

σ12 ≤ σ2

2

σ12 > σ2

2

0.10

9 20

0 F1.96

0.10

Because 400 > 256, 2 21 2400 and 256s s

21

22

4001.56

256

sF

s

There is not enough evidence to convince the manager to switch to the new system.

1.961.56

Fail to Reject H0

Example: Performing a Two-Example: Performing a Two-Sample Sample FF-Test-Test

You want to purchase stock in a company and are deciding between two different stocks. Because a stock’s risk can be associated with the standard deviation of its daily closing prices, you randomly select samples of the daily closing prices for each stock to obtain the results. At α = 0.05, can you conclude that one of the two stocks is a riskier investment? Assume the stock closing prices are normally distributed.


Stock A Stock Bn2 = 30 n1 = 31

s2 = 3.5 s1 = 5.7

Solution: Performing a Two-Solution: Performing a Two-Sample Sample FF-Test-Test


• H0:

• Ha:

• ½α =

• d.f.N= d.f.D=

• Rejection Region:

• Test Statistic:

• Decision:

σ12 = σ2

2

σ12 ≠ σ2

2

0. 025

30 29

0 F2.09

0.025

Because 5.72 > 3.52, 2 2 2 21 25.7 and 3.5s s

2 21

2 22

5.72.65

3.5

sF

s

There is enough evidence to support the claim that one of the two stocks is a riskier investment.

2.092.65

Reject H0

Section 10.3 SummarySection 10.3 SummaryInterpreted the F-distribution and

used an F-table to find critical values

Performed a two-sample F-test to compare two variances


Section 10.3 Comparing Two Variances Larson/Farber 4th ed1.

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