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

Exam questions on Type I and Type II errors

1. 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
2. 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
3. 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