Tests for Equality of Variances: F-Distribution for normal pop.

1
Tests for Equality of VariancesF-Distribution
for normal pop.

• Given 2 independent random samples
• with nx and ny observations
• from 2 normal populations with
• variances sx and sy, std. dev. sx and
  sy.

sx / sy sy / sx ________
2
2
v1 nx - 1
_____
F
F
V2 ny - 1
2
2
____
has an F-distribution with (nx 1) numerator
d.f. and (ny
1) denominator d.f.
2
Example of F-distribution
3
F- table(separate tables for each a level)

• 5 significance level
• v1
• 1 2 3 4
• 1 161.448 199.500 215.707 224.583
• 2 18.513 19.000
  19.164 19.247
• 3 10.128 9.552
  9.277 9.117
• 4 7.709 6.944
  6.591 6.388

v2
v1 numerator degrees of freedom v2
denominator degrees of freedom
4
Example Test Equality of Variance

• The research staff of Investors Now is interested
  
• in determining if there is a difference in the
• variance of the maturities of AAA-rated
  industrial
• bonds compared to CCC-rated industrial bonds.
• A random sample of 17 AAA-rated bonds had a
• sample variance of 123.35, and a random sample
• of 11 CCC-rated bonds had a variance of 8.02.
• a 0.02

5
Example Equality of Variance
2

• H0 sx sy
• H1 sx sy

2
x ? AAA-rated bonds y ? CCC-rated bonds
2
2
?
If H0 is true, the variances are equal and
cancel out, and disappear from the test
statistic
2
2
sx / sy sy / sx
123.35
2
sx
___
______

15.38
______


8.02
2
2
2
sy
15.38 gt 4.53 Reject H0.
16
F F16,10,0.01 4.53
10
critical value
a .01
6
Testing when population variances are
unknown The population variances must be
estimated from sample data (i.e. we must use
sample variances).
7
Small samples with unknown variances 1)
pool sample variances 2) do not pool
sample variances
8
When to pool variances
9
Pooled variance hypothesis test 1
A supplier offers a trucking company manager a
Goodyear or a house brand tire. The house brand
is supposed to be the same tire, made in the same
plant as the Goodyear. The manager suspects that
the house brand tires are "seconds" that will get
less mileage.
10
He puts 1 Goodyear tire on each of 10 trucks and
1 house brand on each of 8 trucks and observes
the remaining tread depth at 50,000 miles.
Assume tire wear is normally distributed.
Test the null that the house brand has at least
as much remaining tread depth as Goodyear.
11
Remaining tread depth
?

?
?
?
0
H
0
H
G

H
?
?
?
?
0
1
H
G
12
(No Transcript)
13
(No Transcript)
14
C.V. (mH mG)0 tasx x
_
_
H
G
C.V. 0 1.746 (9.195)
- 16.054
Do not reject H0
15
Pooled variance hypothesis test 2
Are men dance instructors paid more than
women?
men women 10,400 10,000
9,800 9,900 9,700 10,000
10,500 10,400 10,600 10,600 10,200
9,900 10,300 10,400
10,000 9,700 10,400 10,300
10,700 10,200
16
raw score test statistic
average salary difference
17
Under the effective null hypothesis
mean
variance
18
A sample of 10 men show an average
salary of 10,260 while a sample of 10 women
show an average salary of 10,140.
raw score space
? 0.01
0
C.V.
standardized space
? 0.01
t
19
pooled standard deviation
20
df 10 10 - 2 18 t 2.552 , one
tail, ? .01
df 10 10 - 2 18 t 2.878, two
tail, ? .01
Results are not significant for one-tail test and
not significant for two-tail test.
21
Unpooled sample variances assumes unequal
population variances
_______________________
One half of the students are assigned study
partners and turn in joint homework, the other
half work alone. The instructor wants to test
whether study partners make any difference in the
score on the final exam.
22
x
70
?
78
,
x
?
2
1
s
s
.
?
16
.
,
?
14
2
5
2
1
n
n
?
12
,
?
11
2
1
23
Assume distribution of test scores is normal. ?
0.1
24
(No Transcript)
25
Finding the degrees of freedom when population
variances are unknown and unequal (Fisher-Behrens
problem) no one knows the correct degrees of
freedom.
26
Satterthwaites approx for df
27
(No Transcript)
28
round down to df 20 results in a higher t
value ? slightly less than 0.1
29
Confidence Interval on m, unpooled variance
C
.
V
.
,
C
.
V
.

L
U
(
)
t
s

m
-
m
_

P
A
/
2
x
x
a
-
0
1
2
_

C
.
,
C
.
0

(-11.05 , 11.05)
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Testing difference in proportions with
unequal population variances(need large
samples to assume normality)
31
p1
p2
???????
32
p1-p2
?????
33
First, assume that population variances are
unequal. For proportions, this implies
that either n1 ??n2 or ?1 ? ?2 .
34
Confidence interval Difference between
proportions
(unpooled variances)
?
?

_
L
,
U
?
p
?
p
t
s
?
p
I
M
?
/
2
p
M
I
Example IBM both manufactures and buys DRAM
chips from U.S. Micron. It wants to estimate
the difference in the proportion of the two chips
that are returned during warranty.
35
95 confidence interval
36
95 confidence interval
?
?

p
s
?
L
,
U
?
p
t
_
M
?
/
2
p
?
p
I

I
M
L
,
U


_

0
.
0058
0
.
0082
1
.
96
0
.
000798
(
)
(
)
-
(- 0.004 , - 0.0008)
37
Hypothesis test Difference between two
proportions (unpooled variances) IBM will stop
buying DRAM from U.S. Micron if difference is
smaller than or equal to 0.002. Test at a
0.05.
_
H0 PI PM gt - 0.002 H1 PI PM lt - 0.002
38
test in raw score space
?
?
C
.
V
.
?
?
?
?
?
t
s
I
M
?
p
?
p
0
I
M
?
.
645
.
002
?
0
1
? .05
C.V. - 0.0033
?
0
.
0058
?
0
.
0082
?
p
p
I
M
do not reject H0
?
?
0
.
0024