 Solution

For observed proportions $y_i$ and fitted proportions ${\hat{p}}_i$, the binomial deviance isA second-order Taylor expansion around $y_i={\hat{p}}_i$ givesthe generalized Pearson chi-squared statistic. Under the fitted binomial model this is approximately ${\chi }_{4-3}^2={\chi }_1^2$. The observed deviance is $1.6102$, whose upper-tail probability is about $0.20$. Since this exceeds $0.05$, there is no significant evidence of overdispersion.