Transcript Aim: How do we test the difference between two variances?

Aim: How do we test the
difference between two
variances?
HW#13: complete two questions
on last slides
Comparing Data
• Not only do we test the difference between
two means (yesterdays aim) but
statisticians are interested in comparing t
two variances and standard deviations
What test do we use?
• For comparison of two variances or
standard deviations, an F test is used
Characteristics of the F Distribution
1. The values of F cannot be negative, because
variances are always positive or zero
2. The distribution is positively skewed
3. The mean value of F is approximately equal to
1
4. The F distribution is a family of curves based
on the degrees of freedom of the variance of
the numerator and the degrees of freedom of
the variance of the denominator
Shapes of F Distributions
Formula for the F Test
2
1
2
2
s
F
s
• The larger of the two variances is placed
in the numerator regardless of the
subscripts
Degrees of Freedom
• d.f.N  degree of freedom for numerator
• d.f.D  degree of freedom for
denominator
• NEED TO USE H TABLE
Example
• Find the critical value for a right-tailed F test when
α=0.05, the degrees of freedom for the numerator are
15, and the degrees of freedom for the denominator
are 21.
• Since the test is right tailed with α=0.05, use the 0.05
table. The d.f.N is listed across the top and the d.f.D is
listed in the left column
– Solution: 2.18
Using F table
• When the degree of freedom values
cannot be found in the table, closest
values on the smaller side should be used
– Example: If d.f.N = 14, this value is between
given table values of 12 and 15; therefore 12
should be used, to be the safe side
Testing the equality of two
variances
Left-Tailed
Right-Tailed
H0 :   
2
2
H1 :   
2
2
2
1
2
1
H 0 :  12   22
H1 :  12   22
Two-Tailed
H 0 :  12   22
H1 :  12   22
Steps
1.
2.
3.
4.
5.
State the hypothesis and identify claim
Find the critical value
Compare the test values
Make the decision
Summarize the results
Example
• A medical researcher wishes to see whether
the variance of the heart rates (in beats per
minute) of smokers is different from the
variance of heart rates of people, who do not
smoke. Two samples are selected, and the
data are as shown. Using α=0.05, is there
enough evidence to support the claim?
Smoker
Nonsmoker
n = 26
n = 18
s2 = 36
s2 = 10
Solution
H 0 :  12   22
H1 :  12   22
Critical value from table = 2.56
F = 36/10 = 3.6
3.6 > 2.56
Summarize the results: There is enough evidence to
support the claim that the variance of the heart rates of
smokers and nonsmokers is different
Class Work #2
1.
Using the table H, find the P-value interval for each FTest value
1.
2.
2.
F = 2.97, d.f.N=9, d.f.D=14, right tailed
F=3.32, d.f.N=6, d.f.D=12, two tailed
A researcher claims that the standard deviation of the
ages of cats is smaller than the standard deviation of
the ages of dogs who are owned by families in a large
city. A randomly selected sample of 29 cats has a
standard deviation of 2.7 years and a random sample
of 16 dogs has a standard deviation of 3.5 years. Is the
researcher correct? Use α=0.05. If there is a
difference, suggest a reason for the difference.
Homework
Question 1
Homework
Question 2