# Hypothesis Testing

## Hypothesis Testing Revision

**Hypothesis Testing**

A** hypothesis test** uses some sample data to test whether a **hypothesis** (a belief about the distribution of a random variable) is true. **Hypothesis testing** comes with a substantial amount of terminology.

**Terminology for Hypothesis Testing**

**Hypothesis tests**are based on two**hypotheses**. The**null hypothesis**, H_{0}, is a statement about the**value of a population parameter**(a parameter of the distribution of a random variable) which our data will tell us**whether or not to reject**. The**alternate hypothesis**, H_{1}, is**what we believe the parameter is**if we reject the null hypothesis.

- In general, the
**null hypothesis**is something that**we want to show is false**. This is because hypothesis tests**do not show if something is true**, only if something is false, but we can find out something true that we want to know by showing that the**null hypothesis**is false. To this end, the**null hypothesis is usually that a parameter takes a specific value**, while the**alternate hypothesis is usually that the parameter differs from the specific value**, rather than specifying another value.

- A
**hypothesis test**is the means by which we generate a**test statistic**that directs us to**either reject or not reject**the null hypothesis.

- The
**test statistic is a “summary” of the collected data**, and should have a sampling distribution specified by the**null hypothesis**.

**One or Two Tailed Tests**

**Hypothesis tests** can be **one tailed or two tailed**. This depends on H_{1}.

- In a
**one tailed test**, H_{1} takes the form p>x or p<x, where H_{0} is that p=x. - In a
**two tailed test**, H_{1} takes the form p\neq x, where H_{0} is that p=x.

**Significant Data**

We reject the **null hypothesis** when the data we observe is **unlikely to have occurred** if it were true.

Specifically, we state a **significance level** \alpha before we perform the** hypothesis test**, and if the probability of getting the data we got is less than \alpha in a **one tail test** or less than \dfrac{\alpha}{2} in a** two tail test** if we assume H_{0} is true, then we **reject** H_{0}.

The way we **test the probability** of getting the data is by looking at the **sampling distribution of the test statistic**, which is set by the **null hypothesis**.

You will **usually be told what significance level to use**. Common significance levels include 5\%\;(\alpha)=0.05 and 1\%\;(\alpha)=0.01

**Critical Region**

The **critical region** is the set of values of the **test statistic** that would cause H_{0} to be rejected. The **first value inside the critical region** is called the **critical value**. If the **test statistic** is **as extreme or more extreme** than the critical value, then we **reject** H_{0}.

A **one tailed test has a single critical region**, containing the highest or lowest values. A **two tailed test has two critical regions**, one containing high values and one containing low values.

You can **test whether your data is significant** by finding the critical region and seeing if the **test statistic** falls within it.

**Actual Significance Level and** **p-value**

The **p-value** is the probability of obtaining the results we got if H_{0} is true. If the **p-value** is less than \alpha (or \dfrac{\alpha}{2} for two tail test) we reject H_{0}. If the **p-value** is greater than \alpha (or \dfrac{\alpha}{2} for two tail test) we do not reject H_{0}.

The **actual significance level** is the probability of the data being in the critical region if H_{0} is true. **For continuous data, this is the same as the significance level.** However,** for discrete data, it might differ.**

**Example 1: Binomial Hypothesis Test**

X\sim B(10,p) and we observe x=6

Test, at the 5\% **significance level**, if p is larger than 0.4

**[4 marks]**

H_{0}: p=0.4

H_{1}: p>0.4

\alpha=0.05

\mathbb{P}(X\geq 6)=0.1662>0.05

So we are more likely than the significance level to get the data we observe.

Hence, do not reject H_{0}. There is not significant evidence to suggest p>0.4.

**(Note: More on binomial hypothesis testing can be found in the section Binomial Distribution Hypothesis Tests)**

**Example 2: Normal Hypothesis Test**

The amount by which the train Nicola takes to work is delayed is normally distributed. Observations over a number of years show this delay has a mean of five minutes and a standard deviation of two minutes. Nicola believes this has changed. If she is delayed by ten minutes on Friday, is she right at the 5\% **significance level**?

**[6 marks]**

X\sim N(\mu,2)

H_{0}: \mu=5

H_{1}: \mu\neq 5

Significance level \alpha=0.05

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

Observed data: x=10

\mathbb{P}(X\geq 10)=0.0062<0.025Reject H_{0}. There is sufficient evidence to suggest that the average delay has changed.

**(Note: More on normal hypothesis testing can be found in the section Normal Distribution Hypothesis Tests)**

## Hypothesis Testing Example Questions

**Question 1: **Jane has a die that she believes is biased towards rolling 6. Give hypotheses that could be used to test this.

**[2 marks]**

H_{0}:\mathbb{P}(X=6)=\dfrac{1}{6}

H_{1}:\mathbb{P}(X=6)>\dfrac{1}{6}

**Question 2: **Is the following a one tail or two tail hypothesis test?

H_{0}: p=0.5

H_{1}: p\neq 0.5

**[1 mark]**

The alternate hypothesis is p\neq 0.5 so this is a two tail test.

**Question 3:** How close to the bullseye a dart lands in the dartboard when thrown by Phil has a normal distribution. Phil believes his throws have a mean distance from the bullseye of 1cm, with a standard deviation of 0.4cm. He throws a dart and it lands 1.5cm from the centre. Test, at the 5\% significance level, if he is as good as he says.

**[6 marks]**

X\sim N(\mu,0.4)

H_{0}: \mu=1

H_{1}: \mu>1

Significance level \alpha=5\%

Observed data: x=1.5

\mathbb{P}(X\geq 1.5)=0.1056>0.05Do not reject H_{0}. Insufficient evidence to suggest Phil is not as good at darts as he says.

**Question 4: **Consider the hypothesis test on X\sim B(20,p).

H_{0}: p=0.75

H_{1}: p\leq 0.75

Take the significance level to be 0.05

i) What is the critical region for this test?

ii) What is the actual significance level of this test?

**[4 marks]**

i) The critical region is the region for x for which we reject H_{0}.

Critical region is such that \mathbb{P}(X\leq x)<0.05

In this case, \mathbb{P}(X\leq 12)=0.1018>0.05 while \mathbb{P}(X\leq 11)=0.0409<0.05

So the critical region must be the values 11 and under.

ii) The actual significance is the probability of landing in the critical region so it is \mathbb{P}(X\leq 11)=0.0409

**Question 5: **Marsha notices that her neighbourhood seems to contain far more blue cars than would be normal. She finds a statistic online that says nationally, around 4\% of cars are blue. She then observes 50 cars near her house and 5 of them are blue.

Construct a hypothesis test for this at the 5\% level and find whether or not Marsha is right that her neighbourhood contains more blue cars.

**[6 marks]**

X\sim B(50,0.04)

H_{0}: p=0.04

H_{1}: p>0.04

Significance level \alpha=5\%

Observed data: x=5

\mathbb{P}(X\geq 4)=0.0489<0.05Reject H_{0}. There is sufficient evidence to suggest that there are more blue cars in Marsha’s neighbourhood.

## You May Also Like...

### MME Learning Portal

Online exams, practice questions and revision videos for every GCSE level 9-1 topic! No fees, no trial period, just totally free access to the UK’s best GCSE maths revision platform.