added proof "bin-test"

Author: JoramSoch

P/bin-test.md

@@ -0,0 +1,101 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2023-12-16 20:01:14
+
+title: "Binomial test"
+chapter: "Statistical Models"
+section: "Count data"
+topic: "Binomial observations"
+theorem: "Binomial test"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2023
+    title: "Binomial test"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2023-12-16"
+    url: "https://en.wikipedia.org/wiki/Binomial_test#Usage"
+  - authors: "Wikipedia"
+    year: 2023
+    title: "Binomialtest"
+    in: "Wikipedia – Die freie Enzyklopädie"
+    pages: "retrieved on 2023-12-16"
+    url: "https://de.wikipedia.org/wiki/Binomialtest#Signifikanzniveau_und_kritische_Werte"
+
+proof_id: "P429"
+shortcut: "bin-test"
+username: "JoramSoch"
+---
+
+
+**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](/D/bin):
+
+$$ \label{eq:Bin}
+y \sim \mathrm{Bin}(n,p) \; .
+$$
+
+Then, the [null hypothesis](/D/h0)
+
+$$ \label{eq:bin-test-h0}
+H_0: \; p = p_0
+$$
+
+is [rejected](/D/test) at [significance level](/D/alpha) $\alpha$, 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](/P/bin-pmf):
+
+$$ \label{eq:bin-pmf}
+\mathrm{Bin}(x; n, p) = {n \choose x} \, p^x \, (1-p)^{n-x} \; .
+$$
+
+
+**Proof:** The [alternative hypothesis](/D/h1) relative to $H_0$ for a [two-sided test](/D/test-tail) is
+
+$$ \label{eq:bin-test-h1}
+H_1: \; p \neq p_0 \; .
+$$
+
+We can use $y$ as a [test statistic](/D/tstat). Its [sampling distribution](/D/dist-samp) is given by \eqref{eq:Bin}. The [cumulative distribution function](/D/cdf) (CDF) of the test statistic under the null hypothesis is thus equal to the [cumulative distribution function of a binomial distribution](/P/bin-cdf) with [success probability](/D/bin) $p_0$:
+
+$$ \label{eq:y-cdf}
+\mathrm{Pr}(y \leq z \vert H_0) = \sum_{x=0}^{z} \mathrm{Bin}(x; n, p_0) = \sum_{x=0}^{z} {n \choose x} \, p_0^x \, (1-p_0)^{n-x} \; .
+$$
+
+The [critical value](/D/cval) 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](/D/cval), if the [observed test statistic](/D/test) 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.