Section 10.3 Comparing Two Variances. Section 10.3 Objectives Interpret the F-distribution and use an F-table to find critical values Perform a two-sample.

Section 10.3

Comparing Two Variances

Section 10.3 Objectives

• Interpret the F-distribution and use an F-table to find critical values

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

F-Distribution

• Let 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.

2 21 2 and s s

2 21 2 and σ σ

2122

sF

s

Properties of the F-Distribution

1. 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.

Properties of the F-Distribution

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

F-Distributions

Finding Critical Values for the F-Distribution

1. Specify the level of significance α.

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 α F-table.

b. two-tailed, use the ½α F-table.

Example: Finding Critical F-Values

Find the critical F-value for a right-tailed test when α = 0.10, d.f.N = 5 and d.f.D = 28.

The critical value is F0 = 2.06.

Solution:

Example: Finding Critical F-Values

Find 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 F-Values

½α = 0.025, d.f.N = 4 and d.f.D = 8

The critical value is F0 = 5.05.

Two-Sample F-Test for Variances

To 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.


• Test Statistic2122

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

Finding F-statistic

• Larger variance is always in numerator

• Find: F and dfN and dfD for the following

• a) from samples:

= 842, n1 = 11; = 834, n2 = 18

which variance is larger?

Finding F-statistic


a) from samples:

= 842, n1 = 11; = 834, n2 = 18


Sample 1 (842 > 834)

so, F = dfN = 11-1 = 10 and dfD = 18-1=17

Finding F-statistic


b) from samples:

= 365, n1 = 15; = 402, n2 = 9


Sample 2 (402 > 365)

so, F = dfN = 9-1 = 8 and dfD = 15-1=14


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

2. Specify the level of significance.

3. Determine the degrees of freedom.

4. Determine the critical value.

State H0 and Ha.

Identify α.

Use Table 7 in Appendix B.

d.f.N = n1 – 1 d.f.D = n2 – 1

In Words In Symbols


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.

2122

sF

s

In Words In Symbols

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

• H0:

• Ha:

• α =

• d.f.N= d.f.D=

• Rejection Region:

• Test Statistic:

• Decision:

σ12 ≤ σ2

2

σ12 > σ2

2 (Claim)

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 at the 10% level of significance to convince the manager to switch to the new system.

1.961.56

Fail to Reject H0

Example: Performing a Two-Sample F-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 = 31s2 = 3.5 s1 = 5.7

Solution: Performing a Two-Sample F-Test

• H0:

• Ha:

• ½α =

• d.f.N= d.f.D=

• Rejection Region:

• Test Statistic:

• Decision:

σ12 = σ2

2

σ12 ≠ σ2

2 (Claim)

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

5.72.652

3.5

sF

s

There is enough evidence at the 5% level of significance to support the claim that one of the two stocks is a riskier investment.

2.09 2.652

Reject H0

Section 10.3 Summary

• Interpreted 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. Section 10.3 Objectives Interpret the F-distribution and use an F-table to find critical values Perform a two-sample.

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