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

1. 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 $X\geq8$.

   (a)

   Find the size of the test.

   [1 mark]

   • A$0.0107$
   • B$0.0439$
   • C$0.9453$
   • D$0.0547$

   (b)

   Find the probability of a Type II error when the true value is $p=0.7$.

   [1 mark]

   • A$0.617$
   • B$0.383$
   • C$0.0547$
   • D$0.945$

   (c)

   Find the power of the test when $p=0.6$.

   [2 marks]

   Total for question 1: 4 marks

2. 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.

   (a)

   Find the size of the test.

   [1 mark]

   • A$0.9473$
   • B$0.0527$
   • C$0.0166$
   • D$0.0902$

   (b)

   Find the probability of a Type II error when the true mean is $\lambda=4$.

   [1 mark]

   • A$0.371$
   • B$0.785$
   • C$0.629$
   • D$0.434$

   (c)

   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]

   Total for question 2: 4 marks

3. 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 $\bar x<499$.

   (a)

   Find the size of the test.

   [3 marks]

   (b)

   Find the power of the test when the true mean mass is $498.5$, and explain what this value represents.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A manufacturer claims that the proportion of defective bulbs is $p=0.1$. A buyer takes a random sample of 20 bulbs to test $\mathrm{H}_0:p=0.1$ against $\mathrm{H}_1:p>0.1$, where $X$ is the number of defective bulbs in the sample. Test A rejects $\mathrm{H}_0$ if $X\geq5$. Test B rejects $\mathrm{H}_0$ if $X\geq4$.

   (a)

   Find the size of each test, and the probability of a Type II error for each test when $p=0.2$.

   [6 marks]

   (b)

   The buyer wants a test with size at most $0.05$ that has power at least $0.8$ when $p=0.3$. Find the power of each test when $p=0.3$, and evaluate whether either test meets the buyer's requirements. Suggest what the buyer could change.

   [6 marks]

   Total for question 4: 12 marks

End of questions