Section 11.1 Summary

1. A random variable is multinomial if (a) each independent trial has k possible outcomes, $k=2,3,4,\dots ;$ (b) the ith outcome (category) occurs with probability $p_i$, where $i=1,2,\dots ,k;$ and (c) ${\sum }_{i=1}^k{p}_i=1$ (Law of Total Probability).
2. A goodness of fit test is a hypothesis test used to ascertain whether a random variable follows a particular distribution. In a goodness of fit test, the hypotheses are

   $H_0:\text{ The random variable follows a purticular distribution}$

   $H_a:\text{ The random variable does not follow the distribution specified in }{H}_0$.

   Compare the observed frequencies (actual data from the field) with the expected frequencies when $H_0$ is true. If the difference between the observed and expected frequencies is large, reject $H_0$.

3. The ${\chi }^2$ goodness of fit test is performed using (a) the critical-value method or (b) the p-value method.