Question

Consider a distribution \(\mathrm{N}\left(\mu, \sigma^{2}\right)\) where \(\mu\) is known but \(\sigma^{2}\) is not. Devise a method of producing a confidence interval for \(\sigma^{2}\)

Short answer

To find a confidence interval for \(\sigma^2\), compute the sample variance, \(s^2\), and the Chi-squared test statistic, \(\chi^2\), with \(n-1\) degrees of freedom. Find the Chi-squared critical values for a specified confidence level, say \((1-\alpha)100\%\), and construct the confidence interval for \(\sigma^2\) using these critical values, the sample variance, and the sample size. The \((1-\alpha)100\%\) confidence interval for \(\sigma^2\) is given by: \[\frac{(n-1)s^2}{\chi^2_{1-\frac{\alpha}{2}}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{\frac{\alpha}{2}}}\]

Step-by-step solution

Sample Variance

Compute the sample variance, \(s^2\), from the given data. The sample variance is calculated as: \[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2\] where \(n\) is the sample size, \(\bar{x}\) is the sample mean, and \(x_i\) are the individual data points.

Compute the Chi-squared Test Statistic

Once we have the sample variance \(s^2\), we can compute the Chi-squared test statistic, \(\chi^2\), using the following formula: \[\chi^2 = \frac{(n-1)s^2}{\sigma^2}\] Since \(\sigma^2\) is unknown, we cannot compute the exact value of the Chi-squared test statistic. However, we know that \(\chi^2\) follows a Chi-squared distribution with \(n-1\) degrees of freedom.

Find Chi-squared Critical Values

We then need to find the critical values of the Chi-squared distribution for a specified confidence level, say \((1-\alpha)100\%\), where \(\alpha\) is the chosen significance level. Look up the Chi-squared distribution table or use statistical software to find the critical values \(\chi^2_{\frac{\alpha}{2}}\) and \(\chi^2_{1-\frac{\alpha}{2}}\) such that: \[P\left(\chi^2_{\frac{\alpha}{2}}\leq\chi^2\leq\chi^2_{1-\frac{\alpha}{2}}\right) = 1-\alpha\] These critical values correspond to the lower and upper bounds of the Chi-squared distribution for our \((1-\alpha)100\%\) confidence interval.

Construct Confidence Interval for \(\sigma^2\)

Finally, we can construct the confidence interval for the unknown variance \(\sigma^2\) using the critical values found in Step 3, as well as our sample variance \(s^2\) and sample size \(n\). The \((1-\alpha)100\%\) confidence interval for \(\sigma^2\) is given by the following inequalities: \[\frac{(n-1)s^2}{\chi^2_{1-\frac{\alpha}{2}}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{\frac{\alpha}{2}}}\] By solving for \(\sigma^2\) in these inequalities, we obtain the confidence interval for the unknown variance \(\sigma^2\).