- A-Level Maths

# Hypothesis Testing

A **hypothesis** is a statement that has yet to be proved. In statistics, the hypothesis is about the value of a population parameter, and can be tested by carrying out an experiment or taking a sample of the population.

The **test statistic** is the result of the experiment / the statistic generated from the sample.

In order to perform a hypothesis test, **two hypotheses** are required:

The

**null hypothesis, H₀**is the one you assume to be correctThe

**alternative hypothesis, H₁**is the one you are testing for, to see if the assumed parameter is correct or not.

A specific threshold for the probability of the test statistic must also be defined. If the probability of the test statistic is lower than this threshold, there is sufficient evidence to reject H₀. If it is above the threshold, there is insufficient evidence to reject H₀. This threshold is known as the **significance level**, and is typically set at 1, 5 or 10%.

When ending a hypothesis test, you must conclude by saying whether or not there issufficient evidence to reject H₀. Do not say accept or reject H₁

### Critical Regions & Values

If the test statistic falls within the **critical region**, there is sufficient evidence to reject H₀. The **critical value** is the first value to fall inside the critical region. The **acceptance region** is the set of values that are not in the critical region, so there is insufficient evidence to reject H₀.

The **actual significance level** is the probability of incorrectly rejecting the null hypothesis. What this actually means is that:

the actual significance level is the probability of getting the critical value

### One- and Two-Tailed Tests

Hypothesis tests can be **one-tailed** or **two-tailed**. This refers to how many critical regions there are:

For a

**one-tailed**test,**H₁: p < ...**or**H₁: p > ...**and there is only one critical regionFor a

**two-tailed**test,**H₁: p ≠ ...**so there are two critical regions, one on each 'tail'

See the examples below.

## Hypothesis Tests on Binomial Distributions

Often, hypothesis tests are carried out on discrete random variables that are modelled with a binomial distribution.

### One-Tailed Example

A discrete random variable, X, is distributed as B(12, *p*). Officially, X is distributed with a probability of 0.45. However, there is a suspicion that the probability is, in fact, higher. Find, at the 5% significance level, the critical region and actual significance level of the hypothesis test that should be carried out.

Write out the hypotheses & test st