Index: The Book of Statistical ProofsStatistical Models ▷ Count data ▷ Binomial observations ▷ Binomial test

Theorem: Let $y$ be the number of successes resulting from $n$ independent trials with unknown success probability $p$, such that $y$ follows a binomial distribution:

\[\label{eq:Bin} y \sim \mathrm{Bin}(n,p) \; .\]

Then, the null hypothesis

\[\label{eq:bin-test-h0} H_0: \; p = p_0\]

is rejected at significance level $\alpha$, if

\[\label{eq:bin-test-rej} y \leq c_1 \quad \text{or} \quad y \geq c_2\]

where $c_1$ is the largest integer value, such that

\[\label{eq:bin-test-c1} \sum_{x=0}^{c_1} \mathrm{Bin}(x; n, p_0) \leq \frac{\alpha}{2} \; ,\]

and $c_2$ is the smallest integer value, such that

\[\label{eq:bin-test-c2} \sum_{x=c_2}^{n} \mathrm{Bin}(x; n, p_0) \leq \frac{\alpha}{2} \; ,\]

where $\mathrm{Bin}(x; n, p)$ is the probability mass function of the binomial distribution:

\[\label{eq:bin-pmf} \mathrm{Bin}(x; n, p) = {n \choose x} \, p^x \, (1-p)^{n-x} \; .\]

Proof: The alternative hypothesis relative to $H_0$ for a two-sided test is

\[\label{eq:bin-test-h1} H_1: \; p \neq p_0 \; .\]

We can use $y$ as a test statistic. Its sampling distribution is given by \eqref{eq:Bin}. The cumulative distribution function (CDF) of the test statistic under the null hypothesis is thus equal to the cumulative distribution function of a binomial distribution with success probability $p_0$:

\[\label{eq:y-cdf} \mathrm{Pr}(y \leq z \vert H_0) = \sum_{x=0}^{z} \mathrm{Bin}(x; n, p_0) = \sum_{x=0}^{z} {n \choose x} \, p_0^x \, (1-p_0)^{n-x} \; .\]

The critical value is the value of $y$, such that the probability of observing this or more extreme values of the test statistic is equal to or smaller than $\alpha$. Since $H_0$ and $H_1$ define a two-tailed test, we need two critical values $y_1$ and $y_2$ that satisfy

\[\label{eq:y-cvals} \begin{split} \alpha &\geq \mathrm{Pr}(y \in \left\lbrace 0, \ldots, y_1 \right\rbrace \cup \left\lbrace y_2, \ldots, n \right\rbrace \vert H_0) \\ &= \mathrm{Pr}(y \leq y_1 \vert H_0) + \mathrm{Pr}(y \geq y_2 \vert H_0) \\ &= \mathrm{Pr}(y \leq y_1 \vert H_0) + (1-\mathrm{Pr}(y \leq (y_2-1) \vert H_0) \; . \end{split}\]

Given the test statistic’s CDF in \eqref{eq:y-cdf}, this is fulfilled by the values $c_1$ and $c_2$ defined in \eqref{eq:bin-test-c1} and \eqref{eq:bin-test-c2}. Thus, the null hypothesis $H_0$ can be rejected, if the observed test statistic is inside the rejection region:

\[\label{eq:bin-test-rej-qed} y \in \left\lbrace 0, \ldots, c_1 \right\rbrace \cup \left\lbrace c_2, \ldots, n \right\rbrace \; .\]

This is equivalent to \eqref{eq:bin-test-rej} and thus completes the proof.


Metadata: ID: P429 | shortcut: bin-test | author: JoramSoch | date: 2023-12-16, 20:01.