Testing Binomial Models Using Chi-Square Goodness-of-Fit

Apply the chi-square goodness-of-fit procedure to test whether observed count data are consistent with a binomial distribution, both when the success probability $p$ is specified and when it must be estimated from the data.

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Tutorial

Binomial Goodness-of-Fit: Setup and Expected Counts

Suppose we repeat an experiment $N$ times, where each repetition consists of $n$ independent Bernoulli sub-trials and we record the total number of successes. We want to test whether that count follows a binomial distribution $\text{Bin}(n,p)$ .

For each value $i = 0, 1, \ldots, n,$ let $O_i$ denote the observed number of repetitions yielding exactly $i$ successes. Under $H_0\!: X \sim \text{Bin}(n,p),$ the expected count in category $i$ is

E_i = N \cdot P(X = i) = N \binom{n}{i} p^i (1-p)^{n-i}.

For example, with $n=2,$ $p=0.5,$ and $N=40,$

\begin{align*} E_0 &= 40 \cdot (0.5)^2 = 10, \\ E_1 &= 40 \cdot 2(0.5)(0.5) = 20, \\ E_2 &= 40 \cdot (0.5)^2 = 10. \end{align*}

The expected counts always sum to $N,$ since the binomial probabilities sum to $1.$