5.2. Sampling distribution of a sample variance
Like inferences about the population means and population proportions, the population variability may also be point of interest.
In this section we consider inferences for the standard deviation of a population under the assumption that the population distribution is normal.
To
make inferences about
the
natural choice of a statistic is its sample analogue, which is sample
variance
One
simple explanation of using
as
a divisor in the formula for
is that in a random sample of n observations we have n different values of degrees of freedom. But we know that there are only (n -1) different values that can be uniquely defined. In addition can be shown that the expected value of the computed in this way is the population variance.
The population variance and sample variance are related to a probability distribution known as the Chi- square distribution whose form depends
on (n- 1).
Definition:
Let
be
a random sample from a normal population with mean
and standard deviation
.
Then the distribution
is
called the Chi- square distribution with
degrees
of freedom.
Unlike a normal or Student’s t distribution, the probability density curve
of
a
distribution is an asymmetric curve stretching over the positive side
of the line and having a long right tail. The form of the curve
depends on the
value of the degrees of freedom.(Fig. 5.6). In this figure .
Table
3 in Appendix provides the upper
points
of
distributions
for
various
values of
and
the degree of freedom
.
As in the both cases in the t
and
the normal distributions, the upper
point
denotes
the
value
such that the area to the right is
.
The
lower
point
read from the column
in
the table, have area
to
the right.
For
example, the lower 0.05 point is obtained from the table by using the
column.
Whereas the upper 0.05 point is obtained by reading the column
.
(Fig.5.7).
Example:
Find
.
Solution:
The
upper 0.05 point (is read from the column labeled
),
we find that 
.
Example:
Find
.
Solution:
Example:
Assume that from normally distributed population with variance 2.5, sample of 26 observations are selected.
a) What is the probability that sample variance is greater than 2.8?
b) What is the probability that sample variance is less than 2.2?
Solution:
; 
a)
just
greater than 0.10.
b)
greater
than 0.90.
Example:
It is known that students scores on the final exam follow a normal distribution with standard deviation 6.6. A random sample of 25 students is taken.
a) What is the probability that sample standard deviation of scores is greater than 4?
b) What is the probability that sample standard deviation of scores is
less than 8?
Solution:
; 
a)
more
than 0.99.
b)
between
0.90 and 0.95.
Exercises
1. Using the table for the distribution, find
a) The upper 5 % point when n = 9
b) The upper 1 % point when n =18
c) The lower 2.5 % point when n =12
d) The lower 1 % point when n =24.
2. Find the probability of
a)
b)
c)
,
when n
=11
d)
,
when n
=9
3. The monthly salaries of part time working students follow a normal distribution with standard deviation of 3 000 tg. A random sample of 24 observations was taken.
a) Find the probability that sample standard deviation is more than 4 000 tg.
b) What is the probability that sample standard deviation is less than
2 000 tg?
4. A random sample of 10 households was taken. Suppose that the electric bill they pay follow a normal distribution.
a) The probability is 0.1 that sample variance is greater than what percentage of the population variance?
b) Find any pair of numbers, a and b, such that the following statement holds: The probability is 0.90 that the sample variance is between a %
and b % of the population variance.
5. A production process follows a normal distribution. A sample of 15 produced items was selected.
a) The probability is 0.95 that the sample variance is more than what percentage of the population variance?
b) The probability is 0.99 that the sample variance is greater than what percentage of the population variance?
c) Find any pair of numbers, a and b, such that the following statement holds: The probability is 0.95 that the sample variance is between a %
and b % of the population variance.
Answers
1. a) 15.51; b) 33.41; c) 3.82; d) 10.20; 2. a) 0.05; b) 0.025; c) 0.875;
d) 0.875; 3. a) between 0.025 and 0.010; b) between 0.01 and 0.025;
4. a) 1.6311; or 163.11 %; b) a =0.37 and b =1.88; the probability is 0.90 that the sample variance is between 37 % and 188 % of the population variance; 5. a) 46.92% ; b) 33.28 %; c) a =40.21% and b =186.57 %-the probability is 0.95 that the sample variance is between 40.21 % and
186.57 % of the population variance.