Define a Type II error.

Not rejecting H0 when H0 is false.

What is the probability of a Type I error?

The probability that the test statistic is in the critical region when H0 is true.

Another name for the probability of a Type I error?

The actual significance level of the test.

Two-tailed critical region: how do you find P(Type I)?

Add the probabilities of both tails.

How do you choose the critical region at the 5% level?

The largest region with probability at most 0.05 under H0.

Why can the actual significance level be below 5%?

The distribution is discrete, so probabilities jump.

How should an error be described in context?

State what is wrongly concluded and what is actually true.

Type I and Type II errorsAQA A-Level Further Maths: Flashcards

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Define a Type I error.

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Define a Type I error.: Rejecting $\mathrm{H}_0$ when $\mathrm{H}_0$ is true.
Define a Type II error.: Not rejecting $\mathrm{H}_0$ when $\mathrm{H}_0$ is false.
What is the probability of a Type I error?: The probability that the test statistic is in the critical region when $\mathrm{H}_0$ is true.
Which parameter value do you use to find $\mathrm{P}$ (Type I error)?: The value stated in $\mathrm{H}_0$ .
Another name for the probability of a Type I error?: The actual significance level of the test.
$X\sim\mathrm{B}(20,0.3)$ , reject if $X\ge10$ . Which probability gives the Type I error?: $\mathrm{P}(X\ge10)=1-\mathrm{P}(X\le9)=0.0480$
$Y\sim\mathrm{Po}(6)$ , reject if $Y\le2$ . $\mathrm{P}$ (Type I)?: $0.0620$
Two-tailed critical region: how do you find $\mathrm{P}$ (Type I)?: Add the probabilities of both tails.
Effect of enlarging the critical region on $\mathrm{P}$ (Type I)?: It increases.
How do you choose the critical region at the 5% level?: The largest region with probability at most $0.05$ under $\mathrm{H}_0$ .
Why can the actual significance level be below 5%?: The distribution is discrete, so probabilities jump.
How should an error be described in context?: State what is wrongly concluded and what is actually true.
Probability of at least one Type I error in $n$ independent tests?: $1-(1-\alpha)^n$

Exam questions on Type I and Type II errors

A manufacturer claims that 30% of its cereal boxes contain a prize. To test whether the proportion is greater than 30%, a sample of 20 boxes is taken. Let $X$ be the number of boxes in the sample that contain a prize and $p$ the proportion of all boxes that contain a prize. The test is $\mathrm{H}_0:p=0.3$ against $\mathrm{H}_1:p>0.3$ , and $\mathrm{H}_0$ is rejected if $X\ge10$ .
Find the significance level of the test if the critical region were changed to $X\ge9$ .2 marks
The number of calls received by a call centre in an hour is modelled by $Y\sim\mathrm{Po}(\lambda)$ . To test $\mathrm{H}_0:\lambda=6$ against $\mathrm{H}_1:\lambda<6$ , the manager counts the calls in one hour and rejects $\mathrm{H}_0$ if $Y\le2$ .
The manager wants the probability of a Type I error to be at most 5%. Find the largest critical region of the form $Y\le c$ that she can use.2 marks
A seed company claims that the probability that a seed germinates is $0.8$ . A gardener plants 15 seeds and suspects that the true probability is lower. Let $X$ be the number of seeds that germinate and $p$ the probability that a seed germinates. She tests $\mathrm{H}_0:p=0.8$ against $\mathrm{H}_1:p<0.8$ at the 5% significance level, using $X\sim\mathrm{B}(15,p)$ .
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).