Type II errors and powerAQA A-Level Further Maths: Flashcards

What these 13 flashcards ask

Define a Type II error.
Define the power of a test.
Power in terms of P(Type II)?
Which parameter value do you use for P(Type II)?
Which parameter value do you use for P(Type I)?
P(Type II) in terms of the critical region?
Critical region X\ge8: what is the non-rejection region?
Distribution of \bar x when the true mean is \mu?
True value moves further from H0. Effect on power?
Larger sample size. Effect on power?
Larger critical region. Effect on Type I and Type II probabilities?
Why is a specific true value needed for P(Type II)?
Two-tailed normal test: P(Type II)?

Exam questions on Type II errors and power

A spinner lands on red with probability $p$ . To test $\mathrm{H}_0:p=0.5$ against $\mathrm{H}_1:p>0.5$ , it is spun 10 times and $\mathrm{H}_0$ is rejected if it lands on red 8 or more times. Let $X$ be the number of reds, so $X\sim\mathrm{B}(10,p)$ .
Explain why the power of the test is greater when $p=0.9$ than when $p=0.7$ .2 marks
The number of faults in a 5 m length of cable is modelled by $Y\sim\mathrm{Po}(\lambda)$ . To test $\mathrm{H}_0:\lambda=5$ against $\mathrm{H}_1:\lambda>5$ , a single 5 m length is examined and $\mathrm{H}_0$ is rejected if $Y\ge10$ .
Find the probability that the test makes a Type I error.2 marks
The mass of sugar in a bag is normally distributed with standard deviation $4$ g. A test of $\mathrm{H}_0:\mu=500$ against $\mathrm{H}_1:\mu<500$ uses the mean $\bar x$ of a random sample of 16 bags. $\mathrm{H}_0$ is rejected if $\bar x<498$ .
The true mean mass of sugar in the bags is $499.5$ g. Find the probability of a Type II error.3 marks

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).