Section 11.5 Hypothesis Test for one variance
Proceeding in a similar manner, we can also perform hypothesis testing on variances using the \(\chi^2\)-distribution to determine probabilities.
\begin{equation*}
H_0 : \sigma^2 = \sigma_0^2 \\ H_a: \sigma^2 \neq \sigma_0^2
\end{equation*}
or
\begin{equation*}
H_0 : \sigma^2 \ge \sigma_0^2 \\ H_a: \sigma^2 \lt \sigma_0^2
\end{equation*}
or
\begin{equation*}
H_0 : \sigma^2 \le \sigma_0^2 \\ H_a: \sigma^2 \gt \sigma_0^2
\end{equation*}
\begin{equation*}
\text{test statistic} = T = (n-1) \frac{s^2}{\sigma_0^2}
\end{equation*}
For two-tailed, reject if
\begin{equation*}
T \gt \chi_{1-\alpha/2,n-1}^2 \text{ or } T \lt \chi_{\alpha/2,n-1}^2
\end{equation*}
and for one-tailed to the right if
\begin{equation*}
T \gt \chi_{1-\alpha,n-1}^2
\end{equation*}
and for one-tailed to the left if
\begin{equation*}
T \lt \chi_{\alpha,n-1}^2.
\end{equation*}
Technically, for the two-tailed test you could pick T-values so that the total probability sums to \(\alpha\) in any fashion but generally this probability is split evenly between the two tails as noted above.
Use a \(\alpha = 0.01\) significance level to test the claim that \(\sigma = 17\) if the sample statistics include \(n = 11,\) \(\overline{x} = 106,\) and \(s = 23.\)
The test statistic is
The smaller critical number is
The bigger critical number is
What is your conclusion?
- There is not sufficient evidence to warrant the rejection of the claim that the population standard deviation is equal to 17
- There is sufficient evidence to warrant the rejection of the claim that the population standard deviation is equal to 17
