Further Statistics 1: Hypothesis testingEdexcel A-Level Further Maths: Topic test

1. 1
   The number of rescues made by lifeguards at a beach in one week has historically followed a Poisson distribution with mean $2$. After a change in opening hours, the beach manager believes that the mean number of rescues per week has increased. In a randomly chosen week, $6$ rescues were made. The manager tests the belief at the $5\%$ significance level. Let $X$ be the number of rescues in a week.

   (a)

   Which pair of hypotheses should the manager use?

   [1 mark]

   • A$\mathrm{H}_0:\lambda=2,\ \mathrm{H}_1:\lambda>2$
   • B$\mathrm{H}_0:\lambda=2,\ \mathrm{H}_1:\lambda\neq2$
   • C$\mathrm{H}_0:\lambda=6,\ \mathrm{H}_1:\lambda>6$
   • D$\mathrm{H}_0:\lambda=2,\ \mathrm{H}_1:\lambda<2$

   (b)

   Under $\mathrm{H}_0$, what is the p-value of the observation, to 3 significant figures?

   [1 mark]

   • A$0.0120$
   • B$0.983$
   • C$0.0166$
   • D$0.00453$

   (c)

   State the conclusion of the test, in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A game spinner is claimed to land on red with probability $p=0.25$ on each spin, independently of other spins. Anika suspects that the probability is lower. She spins it until it lands on red and counts the number of spins, $X$, up to and including the first red. She tests the claim at the $5\%$ significance level.

   (a)

   Which pair of hypotheses should Anika use?

   [1 mark]

   • A$\mathrm{H}_0:p=0.25,\ \mathrm{H}_1:p>0.25$
   • B$\mathrm{H}_0:p=0.25,\ \mathrm{H}_1:p<0.25$
   • C$\mathrm{H}_0:p<0.25,\ \mathrm{H}_1:p=0.25$
   • D$\mathrm{H}_0:p=0.25,\ \mathrm{H}_1:p\neq0.25$

   (b)

   The first red occurs on the $9$th spin. What is the p-value, to 3 significant figures?

   [1 mark]

   • A$0.0751$
   • B$0.900$
   • C$0.0563$
   • D$0.100$

   (c)

   Given that the first red occurs on the $9$th spin, state the conclusion of the test, in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A school's computer network has historically crashed at a mean rate of $0.8$ per week, with the number of crashes modelled by a Poisson distribution. After a software upgrade, the technician wants to test, at the $5\%$ significance level, whether the mean rate of crashes has decreased. Over a period of $10$ weeks, $3$ crashes occur.

   (a)

   State suitable hypotheses and the distribution of the number of crashes in $10$ weeks if $\mathrm{H}_0$ is true.

   [3 marks]

   (b)

   Carry out the test, stating your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of earthquakes of magnitude at least $5$ recorded in a region in one year has historically followed a Poisson distribution with mean $6$. A seismologist suspects that the mean number per year has increased and plans to carry out tests at the $5\%$ significance level.

   (a)

   The seismologist first plans to use the number of earthquakes, $X$, in a single year. (i) State suitable hypotheses. (ii) Find the critical region for the test. (iii) State the actual significance level of the test. Marks: (i) 2, (ii) 3, (iii) 1.

   [6 marks]

   (b)

   Over a two-year period, $19$ earthquakes of this kind were recorded. Test, at the $5\%$ significance level, whether the mean number per year has increased.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The number of patients admitted to a hospital ward on a night follows a Poisson distribution with mean $4$ historically. During a flu outbreak, a doctor suspects that the mean has increased. On a randomly chosen night during the outbreak, $8$ patients are admitted. The doctor tests the suspicion at the $5\%$ significance level. Let $X$ be the number of patients admitted on a night.

   (a)

   What is the p-value of the observation, to 3 significant figures?

   [1 mark]

   • A$0.0511$
   • B$0.0298$
   • C$0.949$
   • D$0.111$

   (b)

   What is the correct conclusion of the test?

   [1 mark]

   • AReject $\mathrm{H}_0$: there is evidence that the mean has increased.
   • BReject $\mathrm{H}_0$ because $8$ is greater than $4$.
   • CDo not reject $\mathrm{H}_0$: this proves that the mean is still $4$.
   • DDo not reject $\mathrm{H}_0$: there is insufficient evidence that the mean has increased.

   (c)

   The doctor had no prior suspicion and instead tests $\mathrm{H}_0:\lambda=4$ against $\mathrm{H}_1:\lambda\neq4$ at the $5\%$ level, using the same observation. State the value the p-value must be compared with, and the conclusion.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A bakery claims that each loaf it bakes fails inspection with probability $0.2$, independently of other loaves. An inspector suspects that the true probability is lower. She checks loaves in the order in which they were baked and finds that the first loaf to fail is the $15$th loaf. Let $X$ be the number of loaves checked up to and including the first failure. She tests the claim at the $5\%$ significance level.

   (a)

   What is the p-value of the observation, to 3 significant figures?

   [1 mark]

   • A$0.0352$
   • B$0.0440$
   • C$0.956$
   • D$0.00880$

   (b)

   What is the correct conclusion of the test?

   [1 mark]

   • ADo not reject $\mathrm{H}_0$ because $0.0440<0.05$.
   • BReject $\mathrm{H}_0$ because $0.0440>0.05$.
   • CReject $\mathrm{H}_0$: there is evidence that the probability of a loaf failing inspection is less than $0.2$.
   • DDo not reject $\mathrm{H}_0$: there is evidence that the probability is exactly $0.2$.

   (c)

   Find the critical region for $X$ in this test.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A footballer claims that the goalkeeper saves each of his penalty kicks with probability $0.2$, independently for each kick. A coach thinks the probability is different. The coach records the number of penalties, $X$, taken up to and including the first one that is saved, and tests the claim at the $5\%$ significance level. The first saved penalty is the $19$th penalty.

   (a)

   State suitable hypotheses and find the probability $\mathrm{P}(X\geq19)$ assuming the claim is true.

   [3 marks]

   (b)

   Complete the test, stating your conclusion in context.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Flaws in a cable occur randomly. A supplier claims that the number of flaws in a $1$ m length of its cable follows a Poisson distribution with mean $0.5$. An inspector examines successive $1$ m lengths. Let $X$ be the number of lengths examined up to and including the first length that has at least one flaw.

   (a)

   Assume the supplier's claim is true. (i) Show that the probability that a $1$ m length has at least one flaw is $0.393$ to 3 significant figures. (ii) State the distribution of $X$. (iii) Find $\mathrm{E}(X)$. (iv) Find $\mathrm{P}(X>5)$. Marks: (i) 2, (ii) 1, (iii) 1, (iv) 2.

   [6 marks]

   (b)

   A rival engineer claims that the cable has fewer flaws than the supplier states. The inspector finds that the first length with a flaw is the $9$th length examined. Test the engineer's claim at the $5\%$ significance level, using the value of $p=1-\mathrm{e}^{-0.5}$ from part (a).

   [6 marks]

   Total for question 8: 12 marks

End of questions