8. Confidence Intervals

8.5 Chi Squared Distribution

The \chi^{2} (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 :

  1. Suppose you have a population that has variance \sigma^{2} and is normally distributed.
  2. Take a sample of size n from the population and compute x_{1} =\frac{(n-1)s_{1}^{2}}{\sigma^{2}} using the sample standard deviation s_{1} from that sample.
  3. Put the sample back into the population.
  4. Take another sample of size n from the population and compute x_{2} = \frac{(n-1)s_{2}^{2}}{\sigma} using the sample standard deviation s_{2} from that sample.
  5. etc.
  6. The distribution of the values of x_{i} = \frac{(n-1)s_{i}^{2}}{\sigma^{2}} values will be a \chi^{2} distribution with \nu = n-1 degrees of freedom.

Like the t-distributions, the \chi^2 distributions are a family, see Figure 8.10.

Figure 8.10 : The \chi^{2} distributions are enumerated by degrees of freedom.

The \chi^2 distribution underlies why s is the best estimate for \sigma. It mean, or expected value is \nu = n-1 so the expected value of s is \sigma. The expected value of \sum (x - \bar{x})/n in a random sample of size n is not \sigma.

Confidence Intervals on \sigma and \sigma^{2}

The \chi^{2} distribution is already normalized in its definition through including s in its definition. Therefore no z-transforms are needed and we can work directly with a table that gives right tail areas under the \chi^{2} distribution. That table is the Chi-squared Distribution Table, in the Appendix, and it gives values of \chi^2 for given values of area to the right of \chi^2, see Figure 8.11.

Figure 8.11 : The Chi-squared Distribution Table gives \chi^2 associated with given right tail areas.

We’ll need \chi^2_{\rm left} and \chi^2_{\rm right} such that the tail areas are equal and such that the area between them is \cal{C}, see Figure 8.12.

Figure 8.12 : The values \chi^2_{\rm left} and \chi^2_{\rm right} define the confidence region \cal{C}.

Notation : Let’s call the \alpha in the Chi-squared Distribution Table \alpha_{T} and let \chi^2(\alpha_{T}) be the table value that corresponds to \alpha_T. In other words \chi^2(\alpha_{T}) is the \chi^{2} value that corresponds to a right tail area of \alpha_{T}.

So given \cal{C}, the appropriate \chi^{2}_{\rm left} and \chi^{2}_{\rm right} are the following values from the Chi-squared Distribution Table:

    \[\chi^{2}_{\rm right} = \chi^2 \left( \frac{1 - {\cal{C}}}{2} \right)\]

    \[\chi^{2}_{\rm left} = \chi^2 \left( 1- \left[ \frac{1 - {\cal{C}}}{2} \right] \right).\]

Note the symmetry of the Chi-squared Distribution Table. If \chi^{2}_{\rm right} comes from the column 3 columns from the right edge of the table then \chi^{2}_{\rm left} 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 \sigma^2 is given by

    \[\frac{(n-1)s^{2}}{\chi^{2}_{\rm right}} < \sigma^2 < \frac{(n-1)s^2}{\chi^{-2}_{\rm left}}\]

and for \sigma by:

    \[\sqrt{\frac{(n-1)s^2}{\chi^2_{\rm right}}} < \sigma < \sqrt{\frac{(n-1)s^2}{\chi^2_{\rm left}}}\]

Where the \chi^2 distribution with \nu = n-1 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 \sigma and \sigma^2 for the following data

    \[59, 54, 53, 52, 51, 39, 49, 46, 49, 48\]

Solution : Compute, using your calculator :

    \[s^2 = 28.2\]

    \[\nu =n-1 = 9.\]

From the Chi-squared Distribution Table, in the \nu = 9 line, find :

    \[\chi^2_{\rm right} = \chi^2 \left( \frac{1-0.90}{2}\right) = \chi^2(0.05) = 16.919\]


    \[\chi^2_{\rm left} = \chi^2 (1-0.05) = \chi^2(0.95) = 3.325\]


    \begin{align*} \frac{(n-1)s^2}{\chi^2_{\rm right}} &< \sigma^{2} < \frac{(n-1)s^2}{\chi^2_{\rm left}}\\ \frac{9 \cdot 28.2}{16.919} &< \sigma^{2} < \frac{9 \cdot 28.2}{3.325}\\ 15.0 &< \sigma^2 < 76.3 \hspace{1in} \mbox{with 90\% confidence.} \end{align*}

Taking square roots:

    \[3.87 < \sigma < 8.73 \hspace{1in} \mbox{with 90\% confidence.}\]