Binomial Test

Table of contents

Binomial Distribution
1. Probability Mass Function
2. Sum of Binomial Random Variables
Hypothesis Test
1. Null Hypothesis
2. Calculating the p-value

Binomial Distribution

Binomial distribution is a discrete probability distribution that describes the probability of a success in a binomial (yes-no) experiment.

Each binomial experiment is also called a Bernoulli trial:

Each trial has binary outcome: success or failure
Each trial is independent of each other
The probability of success is the same for each trial, denoted as $p$
The number of trials is fixed, denoted as $n$

When probability of success $p$ is:

$p = 0.5$ , the distribution is symmetric
$p < 0.5$ , the distribution is skewed to the left
$p > 0.5$ , the distribution is skewed to the right

Probability Mass Function

Let $X$ be the random variable that represents the number of successes in $n$ trials, and $p$ be the probability of success in each trial.

Then, the probability mass function of $X$ is:

$P r [X = k] = (\binom{n}{k}) p^{k} (1 - p)^{n - k}$

Binomial Coefficient

(\binom{n}{k}) = \frac{n!}{k! (n - k)!}

where $k$ is the number of successes in $n$ trials.

Then we say that $X$ follows a binomial distribution with parameters $n$ and $p$ :

$X \sim Binomial (n, p)$

Sum of Binomial Random Variables

If $X_{1} \sim Binomial (n_{1}, p)$ and $X_{2} \sim Binomial (n_{2}, p)$ ,

$X_{1} + X_{2} \sim Binomial (n_{1} + n_{2}, p)$

Hypothesis Test

Null Hypothesis

Let $P$ be the probability of success in a binomial experiment.

Supose we were testing whether a coin is fair or not.

Then, the null hypothesis would be:

$H_{0} : P = 0.5$

Meaning the chances of getting heads or tails are equal.

Calculating the p-value

When the null hypothesis is true, we can plug in $P = 0.5$ into the probability mass function:

P r [X = k] = (\binom{n}{k}) {0.5}^{n}

Suppose our sample had 21 heads out of 30 trials.

Now we want to calculate our p-value which is the probability of observing an outcome as extreme as the one we observed, in a binomial distribution with $n = 30$ and $p = 0.5$ .

The probability of this extreme case is the sum of the probabilities of getting 21 or more heads or 9 or less heads:

P r [X \geq 21] + P r [X \leq 9] = \sum_{k = 21}^{30} (\binom{30}{k}) {0.5}^{30} + \sum_{k = 0}^{9} (\binom{30}{k}) {0.5}^{30} \approx 0.043

Remember, with two-tailed test, we need to take both extremes into account.

Based on the results, we can reject the null hypothesis as $0.043 < 0.05$ .