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 s12 and s22 represent the sample variances of two
different populations.
• If both populations are normal and the population
variances σ 12 and σ 22 are equal, then the sampling
distribution of
s12
F 2
s2
is called an F-distribution.
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
F
1
2
3
F-Distributions
4
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.
Solution:
The critical value is F0 = 2.06.
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 righttailed critical value.
•You must remember to use the ½α table.
1
1
  (0.05)  0.025
2
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.
Two-Sample F-Test for Variances
• Test Statistic
s12
F 2
s2
where s12 and s22 represent the sample variances with
s12  s22.
• The degrees of freedom for the numerator is
d.f.N = n1 – 1 where n1 is the size of the sample
having variance s12.
• The degrees of freedom for the denominator is
d.f.D = n2 – 1, and n2 is the size of the sample having
variance s22.
Finding F-statistic
• Larger variance is always in numerator
• Find: F and dfN and dfD for the following
• a) from samples:
𝒔𝟐𝟏 = 842, n1 = 11; 𝒔𝟐𝟐 = 834, n2 = 18
which variance is larger?
Finding F-statistic
• Larger variance is always in numerator
a) from samples:
𝒔𝟐𝟏 = 842, n1 = 11; 𝒔𝟐𝟐 = 834, n2 = 18
which variance is larger?
Sample 1 (842 > 834)
so,
𝟖𝟒𝟐
F=
𝟖𝟑𝟒
dfN = 11-1 = 10 and dfD = 18-1=17
Finding F-statistic
• Larger variance is always in numerator
b) from samples:
𝒔𝟐𝟏 = 365, n1 = 15; 𝒔𝟐𝟐 = 402, n2 = 9
which variance is larger?
Sample 2 (402 > 365)
so,
𝟒𝟎𝟐
F=
𝟑𝟔𝟓
dfN = 9-1 = 8 and dfD = 15-1=14
Two-Sample F-Test for Variances
In Words
1. Identify the claim. State the
null and alternative
hypotheses.
In Symbols
State H0 and Ha.
2. Specify the level of
significance.
Identify α.
3. Determine the degrees of
freedom.
d.f.N = n1 – 1
d.f.D = n2 – 1
4. Determine the critical value.
Use Table 7 in
Appendix B.
Two-Sample F-Test for Variances
In Words
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 Symbols
s12
F 2
s2
If F is in the
rejection region,
reject H0.
Otherwise, fail to
reject H0.
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
•
•
•
•
•
Because 400 > 256, s12  400 and s22  256
H0: σ12 ≤ σ22
• Test Statistic:
s12 400
Ha: σ12 > σ22 (Claim)
F 2 
 1.56
256
s2
α = 0.10
d.f.N= 9 d.f.D= 20
• Decision: Fail to Reject H0
There is not enough evidence
Rejection Region:
at the 10% level of
significance to convince the
0.10
manager to switch to the new
system.
0
1.561.96
F
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
n2 = 30
s2 = 3.5
Stock B
n1 = 31
s1 = 5.7
Solution: Performing a Two-Sample F-Test
•
•
•
•
•
Because 5.72 > 3.52, s12  5.72 and s22  3.52
H0: σ12 = σ22
• Test Statistic:
s12 5.7 2
Ha: σ12 ≠ σ22 (Claim)
F  2  2  2.652
s2 3.5
½α = 0. 025
d.f.N= 30 d.f.D= 29
• Decision: Reject H0
There is enough evidence at
Rejection Region:
the 5% level of significance
to support the claim that one
0.025
of the two stocks is a riskier
investment.
0
2.09 2.652 F
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