1. Comparing Two Variances
Section 10.3
Comparing Two Variances

2. 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

3. 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.

4. Properties of the F-Distribution
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
F-distributions are positively skewed.
The total area under each curve of an F-distribution is equal to 1.

5. Properties of the F-Distribution
Chapter 10
Properties of the F-Distribution
F-values are always greater than or equal to 0.
For all F-distributions, the mean value of F is approximately equal to 1.
d.f.N = 1 and d.f.D = 8 F 1 2 3 4
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
F-Distributions
Larson/Farber 4th ed

6. Finding Critical Values for the F-Distribution
Chapter 10
Finding Critical Values for the F-Distribution
Specify the level of significance α.
Determine the degrees of freedom for the numerator, d.f.N.
Determine the degrees of freedom for the denominator, d.f.D.
Use Table 7 in Appendix B to find the critical value. If the hypothesis test is
one-tailed, use the α F-table.
two-tailed, use the ½α F-table.
Larson/Farber 4th ed

7. 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.
Solution:
The critical value is F0 = 2.06.

8. 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.

9. Solution: Finding Critical F-Values
½α = 0.025, d.f.N = 4 and d.f.D = 8
The critical value is F0 = 5.05.

10. Two-Sample F-Test for Variances
To use the two-sample F-test for comparing two population variances, the following must be true.
The samples must be randomly selected.
The samples must be independent.
Each population must have a normal distribution.

11. Two-Sample F-Test for Variances
Test Statistic
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

12. Finding F-statistic Larger variance is always in numerator
Chapter 10
Finding F-statistic
Larger variance is always in numerator
Find: F and dfN and dfD for the following
a) from samples:
𝒔 𝟏 𝟐 = 842, n1 = 11; 𝒔 𝟐 𝟐 = 834, n2 = 18
which variance is larger?
Larson/Farber 5th ed

13. Finding F-statistic Larger variance is always in numerator
Chapter 10
Finding F-statistic
Larger variance is always in numerator
a) from samples:
𝒔 𝟏 𝟐 = 842, n1 = 11; 𝒔 𝟐 𝟐 = 834, n2 = 18
which variance is larger?
Sample 1 (842 > 834)
so, F = 𝟖𝟒𝟐 𝟖𝟑𝟒 dfN = 11-1 = 10 and dfD = 18-1=17
Larson/Farber 5th ed

14. Finding F-statistic Larger variance is always in numerator
Chapter 10
Finding F-statistic
Larger variance is always in numerator
b) from samples:
𝒔 𝟏 𝟐 = 365, n1 = 15; 𝒔 𝟐 𝟐 = 402, n2 = 9
which variance is larger?
Sample 2 (402 > 365)
so, F = 𝟒𝟎𝟐 𝟑𝟔𝟓 dfN = 9-1 = 8 and dfD = 15-1=14
Larson/Farber 5th ed

15. Two-Sample F-Test for Variances
Chapter 10
Two-Sample F-Test for Variances
In Words In Symbols
Identify the claim. State the null and alternative hypotheses.
Specify the level of significance.
Determine the degrees of freedom.
Determine the critical value.
State H0 and Ha.
Identify α.
d.f.N = n1 – 1 d.f.D = n2 – 1
Use Table 7 in Appendix B.
Larson/Farber 4th ed

16. Two-Sample F-Test for Variances
Chapter 10
Two-Sample F-Test for Variances
In Words In Symbols
Determine the rejection region.
Calculate the test statistic.
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If F is in the rejection region, reject H0. Otherwise, fail to reject H0.
Larson/Farber 4th ed

17. 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.

18. Solution: Performing a Two-Sample F-Test
Because 400 > 256,
H0:
Ha:
α =
d.f.N= d.f.D=
Rejection Region:
σ12 ≤ σ22
σ12 > σ22 (Claim)
Test Statistic:
Decision:
0.10 9 20
Fail to Reject H0
There is not enough evidence at the 10% level of significance to convince the manager to switch to the new system. F
1.96
0.10
1.96
1.56

19. 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 B
n2 = 30
n1 = 31
s2 = 3.5
s1 = 5.7

20. Solution: Performing a Two-Sample F-Test
Because 5.72 > 3.52,
H0:
Ha:
½α =
d.f.N= d.f.D=
Rejection Region:
σ12 = σ22
σ12 ≠ σ22 (Claim)
Test Statistic:
Decision:
0. 025 30 29
Reject H0
There is enough evidence at the 5% level of significance to support the claim that one of the two stocks is a riskier investment. F
2.09
0.025
2.09
2.652

21. 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