# Binomial Distribution Hypothesis Tests

## Binomial Distribution Hypothesis Tests Revision

**Binomial Distribution Hypothesis Tests**

We have done a few **binomial hypothesis tests** on an earlier page, **Hypothesis Testing**, but on this page we shall dive deeper. You need to know **when **to do a **binomial hypothesis test** and also **how **to do a **binomial hypothesis test**, as well as other skills such as **finding the critical region**.

Make sure you are happy with the following topics before continuing.

**When to do a Binomial Hypothesis Test**

There are two types of **hypothesis tests** you need to know about: **binomial distribution hypothesis tests** and **normal distribution hypothesis tests**. In **binomial hypothesis tests**, you are testing the **probability parameter** p. In **normal hypothesis tests**, you are testing the **mean parameter** \mu. This gives us a **key difference** that we can use to determine what test to do and when.

Since **binomial hypothesis tests** test a **probability parameter**, words like **probability**,** proportion** and **percentage** are all clues that you should use a **binomial hypothesis test**. The **situation and the context** should also help – if you would **model it with a binomial distribution**, you want to do a **binomial hypothesis test**.

**How to do a Binomial Hypothesis Test**

- Define the
**parameter**in the**context of the question**– for a**binomial hypothesis test**the parameter is p which is**always the probability**of something. - Write down the
**null hypothesis**and the**alternate hypothesis**. - Define the
**test statistic**X in the**context of the question**. - Write down the
**distribution**of X under the**null hypothesis**. - State the
**significance level**\alpha – even though you are likely given it in the question,**not stating it risks losing a mark**. **Test for significance**or find the**critical region**.- Write a
**concluding sentence**, linking the acceptance or rejection of H_{0} to the**context**.

**Critical Region and Actual Significance Level**

The **critical region** is the region for which you reject the **null hypothesis**. For a **binomial distribution**, this is all the numbers x such that \mathbb{P}(X\geq x) or \mathbb{P}(X\leq x) (depending on what test you are doing) is **less than** \alpha.

The **actual significance level** is the **probability of landing in the critical region**. It seems at first surprising that this should differ from the given **significance level**, and indeed **for a normal distribution it does not**. However, because the **binomial distribution is discrete**, there will **not** be an x value where the probability is exactly equal to the significance level, so the **critical region always starts at a value that is less likely than the significance level** to be obtained.

**Example: Binomial Hypothesis Test**

When Edith buys lunch during a work day, there is a probability of 0.6 that the shop has her favourite sandwich in stock. After only one sandwich being in stock in the last five days, she is certain the probability of sandwiches being in stock has decreased. Test, at the 5\% significance level, if Edith is correct.

**[6 marks]**

p is the probability of the sandwiches being in stock on a given day

H_{0}: p=0.6

H_{1}: p<0.6

Test statistic: X is the number of sandwiches in stock over five days.

Under H_{0}: X\sim B(5,0.6)

Significance level: \alpha=0.05

\mathbb{P}(X\leq 1)=0.0870>0.05Do not reject H_{0}. Insufficient evidence to suggest Edith is correct.

## Binomial Distribution Hypothesis Tests Example Questions

**Question 1: **A disease is moving through a population. On Tuesday, it is believed that nationally around 6\% of people have the disease. In the village of Hammerton, 5 out of 200 residents have the disease. Test, at the 5\% significance level if the prevalence of the disease differs in Hammerton compared to the rest of the country.

**[7 marks]**

p is the probability of having the disease.

H_{0}: p=0.06

H_{1}: p\neq 0.06

Test statistic X is the number of people in Hammerton who have the disease.

Under H_{0}: X\sim B(200,0.06)

Significance level: \alpha=0.05

Two tailed test so we are looking for a probability smaller than \dfrac{0.05}{2}=0.025

\mathbb{P}(X\leq 5)=0.0177<0.025Reject H_{0}. Sufficient evidence to suggest the rate is different in Hammerton compared to the rest of the country.

**Question 2:**

X\sim B(10,p).

i) Find the critical region for a 5\% level one tail test on:

H_{0}: p=0.49

H_{1}: p<0.49

ii) Efan observes X=2 while Izzie observes X=1. Do they reach the same conclusion in a hypothesis test?

**[6 marks]**

i) \mathbb{P}(X=2)=0.0621>0.05

\mathbb{P}(X=1)=0.0126<0.05So 2 is not in the critical region while 1 is.

Hence, the critical region is X=0,1

ii) They reach different conclusions.

Efan observes a value that lies outside the critical region, so concludes that H_{0} should not be rejected.

Izzie observes a value that lies inside the critical region, so concludes that H_{0} should be rejected.

**Question 3: **A scientist is observing a colony of 20 bacteria. It is believed that the probability of a bacterium splitting is 0.32. If the scientist observes just 2 splitting bacteria, test at the 1\% significance level if this believed probability is too high.

**[6 marks]**

p is the probability of a bacterium splitting.

H_{0}: p=0.32

H_{1}: p<0.32

Test statistic X is the number of bacteria that split.

Under H_{0}: X\sim B(20,0.32)

Significance level: \alpha=0.01

\mathbb{P}(X\leq 2)=0.0235>0.01Do not reject H_{0}. Insufficient evidence to suggest the probability of splitting is too high.