exactly normal for any sample size
'
'
n
°
the standard deviation of
x
is referred to as the
standard error (s.e.) on the
mean.
0
0.02
0.04
0.06
0.08
0.1
0.12
Probability
mean of x
4
Example 2.3.1
Forty random observations are taken from Poisson distribution, where
X
is the
number of telephone calls made in an evening to a counselling service, i.e.
(
~
4.5
X
Po
. Find the probability that the sample mean exceeds 5.
2.4
Sample Mean
The standard deviation of a statistic used as an estimator of a population parameter is called the
standard error of the estimator.
Example 2.4.1
Suppose that you select a random sample of
25
n
=
observations from a normal
population with mean
8
μ
=
and
0.6
σ
=
.
Find the probability that the sample mean
X
will:
(a)
be less than 7.9
(b)
exceeds 7.9
(c)
lie within 0.1 of the
8
μ
=
2.4.1
Sample Proportion
Consider a sampling problem involving consumer preference or opinion poll; we
are concerned with estimating the proportion
p
of the people in the population
who possess some specific characteristic.
These are practical examples of
binomial experiments,
if the sampling procedure has been conducted in the appropriate manner
.
(i)
If a random sample of
n
observations is selected from a binomial population
with parameter
p
, then sampling distribution of the sample proportion is given
by:
5
ˆ
x
p
n
=
will have:
ˆ
p
p
μ
=
and
ˆ
p
pq
n
σ
=
(ii)
When the sample size is large, the sampling distribution of
ˆ
p
can be
approximated by a normal distribution.
Example 2.4.2
A survey of 313 children, ages 14 to 22, selected from the nation’s top corporate
executives. When asked to identify the best aspect of being privileged in this group,
55% mentioned material and financial gains.
(i)
Describe the sampling distribution of the sample proportion
(ii)
Assume that the population proportion
is 0.5. What is the probability of
observing a sample proportion as large or larger than
ˆ
p
?
Remarks (ii)
°
This tells us that if we were to select a random sample of
313
n
=
observations
from a population with proportion
0.5
p
=
, the probability that the sample
proportion
ˆ
p
would be as large or larger than 0.55 is only 4%.
(i).
Using the correction of continuity, the equivalent to
0.5
±
would be
1
2
n
±
So;
(
(
29
0.55
0.0016
0.5
1.71
0.0436
0.0283
P Z
P Z


≥
=
≥
=
°
When
n
is sufficiently large, the effect of using the correction of continuity is
generally immaterial.
6
2.4.2
Sum or Difference between two sample mean
When independent random samples of size
1
n
and
2
n
observations have been selected
from population with means
1
μ
and
2
μ
, and variances
2
1
σ
and
2
2
σ
respectively; the
sampling distribution of the sum or differences will have the following properties:
(a)
The mean and standard deviation of
(
1
2
x
x
±
(
29
1
2
1
2
x
x
μ
μ
μ
±
=
±
and
(
29
1
2
2
1
2
1
2
x
x
n
n
σ
σ
σ
±
=
+
(b)
If the sampled populations are normally distributed, then the sampling distribution
is exactly normally distributed regardless of the sample size
(c)
If the sampled populations are not normally distributed, then the sampling
distribution is approximately normally distributed when the sample size are large
due to the CLT
Example 2.4.3
:
2