Section 8.4 Summary

1. The ${\chi }^2$ continuous random variable takes values that are never negative, so the ${\chi }^2$ distribution curve starts at 0 and extends indefinitely to the right. Thus, the ${\chi }^2$ curve is right-skewed and not symmetric. There is a different curve for every different degrees of freedom, $n-1$. To find ${\chi }^2$ critical values, we can use either the ${\chi }^2$ table or technology.

2. If the population is normally distributed, we use the ${\chi }^2$ distribution to construct a $100(1-\alpha )\%$ confidence interval for the population variance ${\sigma }^2$, which is given by

   $\text{lower bound}={\displaystyle \frac{(n-1)s^2}{{\chi }_{\alpha \text{/}2}^2}},\text{upper bound}={\displaystyle \frac{(n-1)s^2}{{\chi }_{1-\alpha /2}^2}}$

   where $s^2$ represents the sample variance and ${\chi }_{1-\alpha /2}^2$ and ${\chi }_{\alpha /2}^2$ are the critical values for a ${\chi }^2$ distribution with $n-1$ degrees of freedom. The confidence interval for $s$ is found by taking the square root of these lower and upper bounds.