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

Exam questions on Type II errors and power

1. 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
2. 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
3. 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