8. Confidence Intervals
8.5 Chi Squared Distribution
The (chi squared) distribution is a consequence of a random process based on the normal distribution. It is derived from the normal distribution as the result of the following stochastic process :
- Suppose you have a population that has variance and is normally distributed.
- Take a sample of size from the population and compute using the sample standard deviation from that sample.
- Put the sample back into the population.
- Take another sample of size from the population and compute using the sample standard deviation from that sample.
- The distribution of the values of values will be a distribution with degrees of freedom.
Like the -distributions, the distributions are a family, see Figure 8.10.
The distribution underlies why is the best estimate for . It mean, or expected value is so the expected value of is . The expected value of in a random sample of size is not .
Confidence Intervals on and
The distribution is already normalized in its definition through including in its definition. Therefore no -transforms are needed and we can work directly with a table that gives right tail areas under the distribution. That table is the Chi-squared Distribution Table, in the Appendix, and it gives values of for given values of area to the right of , see Figure 8.11.
We’ll need and such that the tail areas are equal and such that the area between them is , see Figure 8.12.
Notation : Let’s call the in the Chi-squared Distribution Table and let be the table value that corresponds to . In other words is the value that corresponds to a right tail area of .
So given , the appropriate and are the following values from the Chi-squared Distribution Table:
Note the symmetry of the Chi-squared Distribution Table. If comes from the column 3 columns from the right edge of the table then comes from a column 3 columns from the left edge of the table. Only small and large areas appear in the table, there are no intermediate values.
Finally, the confidence interval for is given by
and for by:
Where the distribution with degrees of freedom (giving the line to use in the Chi-squared Distribution Table) is used.
Example 8.5 : Find the 90 confidence interval on and for the following data
Solution : Compute, using your calculator :
From the Chi-squared Distribution Table, in the line, find :
Taking square roots: