Section10.6Interval Estimates - Confidence Interval for \(\sigma^2\)
Once again, you may need to approximate the population variance or standard deviation but only have the sample values available. One difference from the previous sections is that you are not dealing with an average of values (such as \(\overline{x}\) or \(\tilde{p}\)) but with the average of the squares of values. The Central Limit Theorem does not directly help you in this case but the following result (presented without proof) provides a solution.
Theorem10.6.1.Relationship between Variance and \(\chi ^2\).
If \(S^2\) is a random variable of possible sample variance values from a sample of size n, then
\begin{equation*}
W = \frac{(n-1)S^2}{\sigma^2}
\end{equation*}
is approximately \(\chi ^2(n-1).\)
To create a confidence interval for \(\sigma^2\) first consider an interval of the form
and determine values for the boundaries so that the likelihood of this being true is high. For this case, since the chi-square distribution only has a positive domain and is not symmetrical, you will not expect to determine a symmetrical confidence interval. Therefore, consider
Using the previous theorem, note that the inside variable can be replaced with a chi-square variable. If F is the distribution function for chi-square, then you get
\begin{equation*}
F \left ( \frac{(n-1)S^2}{E_1} \right ) - F \left ( \frac{(n-1)S^2}{E_2} \right ) = 1 - \alpha .
\end{equation*}
For a given value of \(\alpha\) there are many possible choices but often one often utilized is one in which
To determine appropriate values for \(\chi^2_{\alpha/2} \) and \(\chi^2_{1-\alpha/2} \) with equal probabilities in each tail, consider using the interactive cell below:
Given the data 570, 561, 546, 540, 609, 580, 550, 577, 585, determine a 95% confidence interval for \(\sigma^2\text{.}\)
Using the computational forumaula (or your calculator) gives \(s^2 \approx 479.5\text{.}\) Also, notice for n=9, the resulting interval will use a Chi-square variable with 8 degrees of freedom. Using the symmetric option, gives \(\chi_{0.025}^2 = 2.18\) and \(\chi_{0.975}^2 = 17.53\text{.}\) Therefore
Notice, this interval is relatively wide which is a result both of the number of data values being relatively small (n=9) and the actual data values being relatively large and spread out.
Find the critical values \(\chi_L^2 = \chi_{1-\alpha/2}^2\) and \(\chi_R^2 = \chi_{\alpha/2}^2\) that correspond to \(99\)% degree of confidence and the sample size \(n = 22.\)
Suppose that you have n=400 data values and suppose you have computed from those a sample variance of \(s^2 = 479.5\text{.}\) Then, the only change in the calculation is the two chi-square statistic values. For 95% but now with 399 degrees of freedom \(\chi_{0.025}^2 = 345.55\) and \(\chi_{0.975}^2 = 456.24\text{.}\)
which is a relatively tight confidence interval. Notice, these are also completely contained in the confidence intervals from the previous small n example.
Similar to above, another choice to estimate \(\sigma ^2\) is to use a one sided confidence interval. If you want to find one of these, continue as described above but just leave one endpoint off. Indeed,
Finally, to determine a confidence interval for \(\sigma\text{,}\) proceed using the protocols described above and simply take the square root on the resulting interval.