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.

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