 Solution

Pearson's statistic isFor independent observations under the independence null hypothesis, its large-sample null distribution is approximately ${\chi }^2$ with $(4-1)(3-1)=6$ statistical degrees of freedom. The fitted row and column probabilities account for the reduction in statistical degrees of freedom. All expected counts exceed five, so the usual approximation has adequate expected cell sizes here. Reject independence when the statistic exceeds the corresponding upper-tail critical value.