Section 9.6 Summary

1. Under the assumption that $H_0$ is true, the ${\chi }^2$ statistic takes the following form:${\chi }_{\text{data}}^2=\frac{\left(n-1\right)s^2}{{\sigma }_0^2}$

   The hypothesis test about $\sigma$ may be performed using the $p$-value method or the critical-value method. Either way, the test is valid only if we have a random sample from a normal population. The critical-value method compares ${\chi }_{\text{data}}^2$ with one or two critical values.

2. The $p$-value method compares the $p$-value to the level of significance $\alpha$.

3. If we have a $100\left(1-\alpha \right)\%$ confidence interval for $\sigma$, of the form (lower bound, upper bound), and are interested in a two-tailed hypothesis test, using $\alpha$, of the form

   $\begin{array}{lll}{H}_0:\sigma ={\sigma }_0\hfill & \text{versus}\hfill & H_a:\sigma \ne {\sigma }_0\hfill \end{array}$

   we will not reject $H_0$ for values of ${\sigma }_0$ that lie between the lower bound and upper bound of the confidence interval. We will reject $H_0$ for values of ${\sigma }_0$ that lie outside this interval.