### Statistical Inference for $\sigma^2$

 $n=$ $s^2=$

for $\sigma^2$:

$H_0:\sigma^2$
$H_a:\sigma^2$

Significance level: $\alpha =$

Show equations

This applet computes performs inference for a population variance $\sigma^2$.

#### Assume

$X_i \stackrel{iid}{\sim} N(\mu,\sigma^2)$ for $i=1,\ldots,n$ ($\sigma^2$ is unknown).
Observed data is $x_1,\ldots,x_n$; sample variance is $s^2$.

#### Notation

$\chi^2_{\alpha/2,n-1}$ is the point on the number line such that $P(\chi^2_{(n-1)} > \chi^2_{\alpha/2,n-1})=\frac{\alpha}{2}$.
$\chi^2_{1-\alpha/2,n-1}$, $\chi^2_{\alpha,n-1}$, and $\chi^2_{1-\alpha,n-1}$ are defined similarly.

#### Directions

• Enter the sample size in the $n$ box.
• Enter the sample variance in the $s^2$ box.
• Hitting "Tab" or "Enter" on your keyboard will compute a 95% confidence interval for $\sigma^2$ (you can change the confidence level using the drop-down box). Upper and lower bounds (one-sided CIs) can be computed as well.

To perform a hypothesis test for $\sigma^2$, enter $H_0$ and $H_a$. Specify the null and alternative hypotheses with the drop-down boxes. The critical value, rejection region, test statistic, and $p-$value are computed and graphed. Different significance levels can be chosen with the drop-down box.