Inference About a Population Variance
• Sometimes we are interested in making inference
about the variability of processes.
• Examples:
– Investors use variance as a measure of risk.
• To draw inference about variability, the parameter
of interest is s2.
1
Inference About a Population Variance
• The sample variance s2 is an unbiased, consistent and
efficient point estimator for s2.
( n  1) s
2
• The statistic
has a distribution called Chi2
s
squared, if the population is normally distributed.

d.f. = 5
2

( n  1) s
s
2
2
d .f .  n  1
d.f. = 10
2
Testing and Estimating a Population
Variance
• From the following probability statement
P(21-a/2 < 2 < 2a/2) = 1-a
we have (by substituting 2 = [(n - 1)s2]/s2.)
( n  1) s
2
a /2
2
2
s 
( n  1) s
2
2
 1 a / 2
3
Testing the Population Variance
• Example
– H0: s2 = 1
H1: s2 <1
The
The
test
statistic
rejection
2
is  
region
( n  1) s
s
2
2
.
2
2
is    1 a , n  1
4
Inference about Two Populations
• Variety of techniques are presented whose
objective is to compare two populations.
• We are interested in:
– The difference between two means.
– The ratio of two variances.
5
13.2 Inference about the Difference
between Two Means: Independent
Samples
• Two random samples are drawn from the two
populations of interest.
• Because we compare two population means, we
use the statistic x 1  x 2.
6
The Sampling Distribution of x  x
1
2
1.
x 1  x 2 is normally distributed if the (original)
population distributions are normal .
2.
x 1  x 2 is approximately normally distributed if
the (original) population is not normal, but the
samples’ size is sufficiently large (greater than 30).
3.
The expected value of x 1  x 2 is m1 - m2
4.
The variance of x 1  x 2 is s12/n1 + s22/n2
7
Making an inference about m – m
• If the sampling distribution of x  x is normal or
approximately normal we can write:
1
Z 
2
( x 1  x 2 )  (m   m  )
s

n1
s


n2
• Z can be used to build a test statistic or a
confidence interval for m1 - m2
8
Making an inference about m – m
• Practically, the “Z” statistic is hardly used,
because the population variances are not known.
Zt 
( x 1  x 2 )  (m   m  )
sS?12

n1
sS?22


n2
• Instead, we construct a t statistic using the
sample “variances” (S12 and S22).
9
Making an inference about m – m
• Two cases are considered when producing the
t-statistic.
– The two unknown population variances are equal.
– The two unknown population variances are not equal.
10
Inference about m – m: Equal variances
• Calculate the pooled variance estimate by:
The pooled
Variance
estimator
( n 1  1) s 1  ( n 2  1) s 2
2
Sp 
2
2
n1  n 2  2
n2 = 15
n1 = 10
2
S1
2
S2
2
Sp
Example: s12 = 25; s22 = 30; n1 = 10; n2 = 15. Then,
Sp 
2
(10  1)( 25 )  (15  1)( 30 )
10  15  2
 28 . 04347
11
Inference about m – m: Equal variances
• Construct the t-statistic as follows:
t
( x 1  x 2 )  (m   m  )
2
p
s (
1
n1

1
n2
)
d .f .  n 1  n 2  2
• Perform a hypothesis test
H0: m  m = 0
H1: m  m > 0
or < 0
or  0
Build a confidence interval
(x1  x2 )  t a 
sp (
2
1
n1

1
n2
)
where   a  is the confidence level.
12
Inference about m – m: Unequal variances
t 
( x1  x 2 )  ( m   m  )
2
(
s1
2

n1
2
d.f. 
( s1
2
( s1
s2
)
n2
2
n1  s 2 / n 2 )
n1 )
n1  1
2

2
(s2
2
2
n2 )
n2  1
13
Inference about m – m: Unequal variances
Conduct a hypothesis test
as needed, or,
build a confidence interval
Confidence
(x  x )  t
1
2
a 2
interval
2
2
s
s
( 1  2 )
n
n
1
2
where   a  is the confidence level
14
Inference about the ratio of two variances
• In this section we draw inference about the ratio
of two population variances.
• This question is interesting because:
– Variances can be used to evaluate the consistency
of processes.
– The relationship between population variances
determines which of the equal-variances or unequalvariances t-test and estimator of the difference
between means should be applied
15
Parameter and Statistic
• Parameter to be tested is s12/s22
• Statistic used is
F
s
2
1
s
2
2
s
2
1
s
2
2
• Sampling distribution of the statistic
The statistic [s12/s12] / [s22/s22] follows the F distribution with
n1 = n1 – 1, and n2 = n2 – 1.
16
Parameter and Statistic
– Our null hypothesis is always
H0: s12 / s22 = 1
S12/s12
– Under this null hypothesis the F statistic F = 2 2
S2 /s2
becomes
F
s
2
1
s
2
2
17