2.1
Two independent samples from two Normal distributions
•
Population/Process 1’s sample
measurements
x
1
, . . . , x
n
1
are a realization of a random
sample of size
n
1
from
N
(
μ
1
, σ
). Let ¯
x
&
s
1
denote the observed sample mean &
standard deviation.
•
Population/Process 2’s sample
measurements
y
1
, . . . , y
n
2
are a realization of a random
sample of size
n
2
from
N
(
μ
2
, σ
). Let ¯
y
&
s
2
denote the observed sample mean &
standard deviation.
•
The two random samples are
independent
.
•
The quantity ¯
x

¯
y
is an estimate of
μ
1

μ
2
.
•
The quantity
s
p
, deFned as:
s
p
=
r
(
n
1

1)
s
2
1
+ (
n
2

1)
s
2
2
n
1
+
n
2

2
is an estimate of
σ
. This is a realization of the estimator
S
p
.
We are assuming that Population/Process 1’s distribution has the same standard deviation
σ
as Population/Process 2’s distribution.
Example 2.1:
Suppose we wish to compare the mean lifetimes (in hours) of two types of batteries (Brand
A and Brand B). We acquire 5 brand A batteries and measure lifetimes in a portable CD
player:
10
.
5
,
9
.
6
,
10
.
2
,
9
.
9
,
10
.
4
We also acquire 6 brand B batteries and measure lifetimes in a portable CD player:
8
.
9
,
9
.
4
,
9
.
7
,
9
.
3
,
8
.
8
,
10
.
0
Assume that brand A battery lifetimes:
x
1
, . . . , x
5
are a realization of a random sample
from
N
(
μ
1
, σ
) and brand B battery lifetimes:
y
1
, . . . , y
6
are a realization of an independent
random sample from
N
(
μ
2
, σ
). Using the data, compute estimates for
μ
1

μ
2
, and
σ
.