17.3.5.2 Algorithms (One Sample Test for Variance)

Let \(\sigma\,\!\) be the variance of sample\(x\,\!\) and \(\sigma_0\,\!\) is the hypothetical variance, this function tests the hypotheses:

\(H_0:\sigma = \sigma_0\,\!\) vs \(H_1:\sigma \ne \sigma_0\,\!\)

\(H_0:\sigma\le\sigma_0\) vs \(H_1:\sigma > \sigma_0\,\!\)

\(H_0:\sigma\ge\sigma_0\) vs \(H_1:\sigma < \sigma_0\,\!\)

Test Statistics

To compare the variance, we firstly compute the Chi-square value by:

\[x^2=\frac{(n-1)s^2}{\sigma_0^2}\]

where \(s^2\,\!\) is the sample variance. For a given significance level, \(\alpha\,\!\), we will reject the null hypothesis \(H_0\,\!\) when:

\(|x^2| \ne \chi_{\alpha/2}^2\,\!\), for two tailed test

\(x^2>\chi_\alpha^2\), for upper tailed test

\(x^2<\chi_{1-\alpha}^2\), for lower tailed test

Confidence Intervals

And the confidence interval for the sample variance can be generated by:

Null Hypothesis	Confidence Interval
\[H_0:\sigma = \sigma_0\,\!\]	\[\sqrt{\frac{(n-1)s^2}{\chi_{(\alpha/2,n-1)}^2}} \le \sigma \le \sqrt{\frac{(n-1)s^2}{\chi_{(1-\alpha/2,n-1)}^2}}\]
\[H_0:\sigma\le\sigma_0\]	\[\sqrt{\frac{(n-1)s^2}{\chi_{(\alpha,n-1)}^2}} \le \sigma \le \infty\]
\[H_0:\sigma\ge\sigma_0\]	\[0 \le \sigma \le \sqrt{\frac{(n-1)s^2}{\chi_{(1-\alpha,n-1)}^2}}\]