What is the null hypothesis H0?

The claim assumed to be true, giving a single value of the parameter, such as p=0.5.

What is the alternative hypothesis H1?

What is accepted if H0 is rejected: pp0, p<p0 or p\ne p0.

What must hypotheses be written in terms of?

A population parameter, such as p or \lambda, never the sample or test statistic.

What is a test statistic?

The quantity calculated from the sample, with a known distribution under H0.

What is the critical region?

The set of values of the test statistic for which H0 is rejected.

What is the significance level?

The probability of rejecting H0 when it is true; the probability assigned to the critical region.

What is a critical value?

The boundary of the critical region.

When is a test one-tailed?

When H1 is in one direction only, for example p0.5 or \lambda<7.5.

How is the significance level used in a two-tailed test?

It is split equally between the tails, so 5\% gives 2.5\% in each tail.

What is the actual significance level?

The true probability of the test statistic being in the critical region. For a discrete variable it is usually smaller than the stated level.

What do you do if the test statistic is in the critical region?

Reject H0 and state in context that there is sufficient evidence for H1.

What do you conclude if the test statistic is not in the critical region?

Do not reject H0: there is insufficient evidence for H1. This does not prove H0.

How can a hypothesis test refine a model?

If H0, the current model, is rejected, the model no longer fits and is changed, for example by using a new parameter value.

Hypothesis testing conceptsEdexcel International A Level Further Maths: Flashcards

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What is a hypothesis test?: A procedure that uses sample data to decide whether there is sufficient evidence to reject a claim about a population parameter.
What is the null hypothesis $\mathrm{H}_0$ ?: The claim assumed to be true, giving a single value of the parameter, such as $p=0.5$ .
What is the alternative hypothesis $\mathrm{H}_1$ ?: What is accepted if $\mathrm{H}_0$ is rejected: $p>p_0$ , $p<p_0$ or $p\ne p_0$ .
What must hypotheses be written in terms of?: A population parameter, such as $p$ or $\lambda$ , never the sample or test statistic.
What is a test statistic?: The quantity calculated from the sample, with a known distribution under $\mathrm{H}_0$ .
What is the critical region?: The set of values of the test statistic for which $\mathrm{H}_0$ is rejected.
What is the significance level?: The probability of rejecting $\mathrm{H}_0$ when it is true; the probability assigned to the critical region.
What is a critical value?: The boundary of the critical region.
When is a test one-tailed?: When $\mathrm{H}_1$ is in one direction only, for example $p>0.5$ or $\lambda<7.5$ .
How is the significance level used in a two-tailed test?: It is split equally between the tails, so $5\%$ gives $2.5\%$ in each tail.
What is the actual significance level?: The true probability of the test statistic being in the critical region. For a discrete variable it is usually smaller than the stated level.
What do you do if the test statistic is in the critical region?: Reject $\mathrm{H}_0$ and state in context that there is sufficient evidence for $\mathrm{H}_1$ .
What do you conclude if the test statistic is not in the critical region?: Do not reject $\mathrm{H}_0$ : there is insufficient evidence for $\mathrm{H}_1$ . This does not prove $\mathrm{H}_0$ .
How can a hypothesis test refine a model?: If $\mathrm{H}_0$ , the current model, is rejected, the model no longer fits and is changed, for example by using a new parameter value.

Exam questions on Hypothesis testing concepts

A coin is suspected of being biased towards heads. It is tossed $20$ times and $X$ is the number of heads. Let $p$ be the probability of heads on one toss.
At the $5\%$ significance level, the critical region is $X\ge15$ . Explain what is meant by the critical region, and state the conclusion if $16$ heads are observed.2 marks
Before a timetable change, the number of late arrivals at an airport gate each day was modelled by a Poisson distribution with mean $4$ . After the change, the manager wants to test whether the mean number $\lambda$ of late arrivals per day is different, using the number $X$ of late arrivals on a randomly chosen day.
The manager says that the test is being used to refine the model. Explain how the test helps to refine the model.2 marks
A teacher gives a pupil $20$ true/false questions to test whether the pupil is just guessing. Let $X$ be the number of questions answered correctly and $p$ the probability that the pupil answers a question correctly. The teacher tests $\mathrm{H}_0:p=0.5$ against $\mathrm{H}_1:p\ne0.5$ at the $5\%$ significance level, with $2.5\%$ in each tail. A calculator may be used.
Find the critical region for the test.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).