Hypothesis testing conceptsEdexcel International A Level Further Maths: Subtopic test

1. 1
   A coin is suspected of being biased towards heads. It is tossed $20$ times and $X$ is the number of heads. Let $p$ be the probability of heads on one toss.

   (a)

   Which pair of hypotheses should be used to test the suspicion?

   [1 mark]

   • A$\mathrm{H}_0:p=0.5,\ \mathrm{H}_1:p\ne0.5$
   • B$\mathrm{H}_0:p>0.5,\ \mathrm{H}_1:p=0.5$
   • C$\mathrm{H}_0:p=0.5,\ \mathrm{H}_1:p<0.5$
   • D$\mathrm{H}_0:p=0.5,\ \mathrm{H}_1:p>0.5$

   (b)

   In this test, the random variable $X$ is called the

   [1 mark]

   • Acritical region
   • Bsignificance level
   • Ctest statistic
   • Dsampling frame

   (c)

   At the $5\%$ significance level, the critical region is $X\ge15$. Explain what is meant by the critical region, and state the conclusion if $16$ heads are observed.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Before a timetable change, the number of late arrivals at an airport gate each day was modelled by a Poisson distribution with mean $4$. After the change, the manager wants to test whether the mean number $\lambda$ of late arrivals per day is different, using the number $X$ of late arrivals on a randomly chosen day.

   (a)

   Which pair of hypotheses should the manager use?

   [1 mark]

   • A$\mathrm{H}_0:\lambda=4,\ \mathrm{H}_1:\lambda>4$
   • B$\mathrm{H}_0:\lambda=4,\ \mathrm{H}_1:\lambda\ne4$
   • C$\mathrm{H}_0:\lambda\ne4,\ \mathrm{H}_1:\lambda=4$
   • D$\mathrm{H}_0:X=4,\ \mathrm{H}_1:X\ne4$

   (b)

   The test is carried out at the $5\%$ significance level. This means that $5\%$ is

   [1 mark]

   • Athe probability that $\mathrm{H}_0$ is true
   • Bthe probability that $\mathrm{H}_1$ is true
   • Cthe probability of rejecting $\mathrm{H}_0$ when $\mathrm{H}_0$ is actually true
   • Dthe probability of not rejecting $\mathrm{H}_0$ when $\mathrm{H}_1$ is true

   (c)

   The manager says that the test is being used to refine the model. Explain how the test helps to refine the model.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A teacher gives a pupil $20$ true/false questions to test whether the pupil is just guessing. Let $X$ be the number of questions answered correctly and $p$ the probability that the pupil answers a question correctly. The teacher tests $\mathrm{H}_0:p=0.5$ against $\mathrm{H}_1:p\ne0.5$ at the $5\%$ significance level, with $2.5\%$ in each tail. A calculator may be used.

   (a)

   Find the critical region for the test.

   [3 marks]

   (b)

   Find the actual significance level of the test. The pupil answers $14$ questions correctly. State the conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A weaver's cloth has faults occurring at random at a mean rate of $2.5$ per $10$ m length. After servicing a machine, the supplier claims that the rate of faults has fallen. To test this, a $30$ m length of cloth is inspected and the number $X$ of faults is recorded. The test is at the $5\%$ significance level. A calculator may be used.

   (a)

   (i) Write down suitable hypotheses, defining the parameter used.
   (ii) Explain why the test is one-tailed.
   (iii) Find the critical region for the test.

   [6 marks]

   (b)

   Exactly $3$ faults are found in the $30$ m length.
   (i) Use the critical region from the $5\%$ test to state the conclusion in context.
   (ii) A colleague proposes a $10\%$ significance level instead. Find the critical region for this test and state the conclusion.
   (iii) Comment on the two conclusions.

   [6 marks]

   Total for question 4: 12 marks

End of questions