# ML Wiki

## Exact Binomial Test

This is a Statistical Test for proportions that uses the Binomial Distribution as the null (sampling) distribution.

It doesn't use the Normal Approximation

• because sometimes it's possible to use the Binomial model directly
• or because it's not possible to use the Normal Model: some conditions are not met

### Binomial Model

Recall the formula:

• $P(\text{success}) = { n \choose k } p^k (1 - p)^{n - k}$
• this is the null distribution of our test

Test

• the tail area of the null distribution:
• add up the probabilities (using the formula) for all $k$ that support the alternative hypothesis $H_A$
• one-sided test - use single tail area
• two-sided - compute single tail and double it

## Examples

### Example 1: Medical Consultant (One-Sample)

• medical consultant helps patients
• he claims that with his help the ratio of complications is lower than usually
• (i.e. lower than 0.10)
• is it true?

We want to test a hypothesis:

• $H_0: p_A = 0.10$ - ratio of complications without a specialist
• $H_A: p_A < 0.10$ - specialist helps, the complications ratio is lower than usual

Observed data:

• 3 complications in 62 cases
• $\hat{p} = 0.048$
• is it only due to chance?

Normal Model

Apply the Binomial Model:

• $p\text{-val} = \sum_{j = 0}^3 { n \choose j } p^j (1 - p)^{n - j} = 0.0015 + 0.01 + 0.034 + 0.0355 = 0.121$
• we don't reject the $H_0$ at $\alpha = 0.05$

check! sim got 0.04