e-Statistics > 4480-5480 Probability and Statistics II

Chi-square Distribution

The chi-square distribution has the number of degrees of freedom (df) = . The curve lies on the positive line, and its shape is skewed to the right particularly when df is small. The value $ x_p$ at the quantile function $ Q(p)$ is numerically obtained as follows:

$ x_p =$ $ \Leftrightarrow$ $ P(X \le x_p) =$

Here $ X$ has a chi-square distribution.

Chi-square distribution table contains the selected quantiles in a useful form of table.

First we obtain the $ \chi^2$-test statistic $ Q =$ . Then we can calculate the p-value

$ P(X > q) =$
so that the value $ Q = q$ belongs to the critical region if it is less than $ \alpha$.