What is a Type II error?

Failing to reject H0 when it is false.

What is the size of a test?

The probability of a Type I error: P(reject H0\midH0 true).

What is the power of a test?

P(reject H0\midH1 true)=1-P(Type II)

What is the power function?

The power of the test written as a function of the true parameter value.

What is the effect of widening the critical region?

The size increases and the power increases (Type II probability decreases).

Distribution of the sample mean for N(\mu,\sigma^2) data, sample size n?

\bar X\simN\left(\mu,\frac{\sigma^2}{n}\right)

Why is the actual size of a discrete test usually below the significance level?

The critical region must consist of whole values, so its probability cannot be tuned exactly.

Type I and Type II errors and powerEdexcel A-Level Further Maths: Flashcards

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What is a Type I error?

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What is a Type I error?: Rejecting $\mathrm{H}_0$ when it is true.
What is a Type II error?: Failing to reject $\mathrm{H}_0$ when it is false.
What is the size of a test?: The probability of a Type I error: $\mathrm{P}(\text{reject }\mathrm{H}_0\mid\mathrm{H}_0\text{ true})$ .
What is the power of a test?: $\mathrm{P}(\text{reject }\mathrm{H}_0\mid\mathrm{H}_1\text{ true})=1-\mathrm{P}(\text{Type II})$
What is the power function?: The power of the test written as a function of the true parameter value.
What value of the parameter do you use to find a Type II error probability?: A specific value from $\mathrm{H}_1$ .
What value do you use to find the size of a test?: The value given in $\mathrm{H}_0$ .
$\mathrm{B}(10,0.5)$ , critical region $X\geq8$ : size?: $\frac{56}{1024}=0.0547$
What happens to the power as the true parameter moves away from the $\mathrm{H}_0$ value?: It increases.
What is the effect of widening the critical region?: The size increases and the power increases (Type II probability decreases).
How can both error probabilities be reduced?: By increasing the sample size.
Distribution of the sample mean for $\mathrm{N}(\mu,\sigma^2)$ data, sample size $n$ ?: $\bar X\sim\mathrm{N}\left(\mu,\frac{\sigma^2}{n}\right)$
Why is the actual size of a discrete test usually below the significance level?: The critical region must consist of whole values, so its probability cannot be tuned exactly.

Exam questions on Type I and Type II errors and power

A coin is suspected of being biased towards heads. It is tossed 10 times and $X$ is the number of heads, where $X\sim\mathrm{B}(10,p)$ . The hypotheses are $\mathrm{H}_0:p=0.5$ and $\mathrm{H}_1:p>0.5$ , and $\mathrm{H}_0$ is rejected if $X\geq8$ .
Find the power of the test when $p=0.6$ .2 marks
The number of defects on a sheet of metal is modelled by $\mathrm{Po}(\lambda)$ . One sheet is inspected to test $\mathrm{H}_0:\lambda=2$ against $\mathrm{H}_1:\lambda>2$ . The test rejects $\mathrm{H}_0$ if the sheet has 5 or more defects.
The critical region is changed to $X\geq6$ . State, with a supporting calculation, the effect on the size of the test and on the power of the test when $\lambda=4$ .2 marks
A machine fills cereal boxes, and the mass of cereal in a box, in grams, is normally distributed with standard deviation $2$ . The mean mass should be $500$ but the manager suspects it is lower. The manager tests $\mathrm{H}_0:\mu=500$ against $\mathrm{H}_1:\mu<500$ using the mean $\bar x$ of a random sample of 16 boxes, and rejects $\mathrm{H}_0$ if $\bar x<499$ .
Find the size of 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).