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

Exam questions on Type I and Type II errors and power

1. 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
2. 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
3. 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