Hypothesis tests for binomial and PoissonEdexcel International A Level Further Maths: Subtopic test

1. 1
   A seed company claims that $70\%$ of its seeds germinate. A gardener plants $20$ seeds and suspects that the true proportion is lower. Let $p$ be the probability that a seed germinates and $X$ the number of the $20$ seeds that germinate. She tests $\mathrm{H}_0:p=0.7$ against $\mathrm{H}_1:p<0.7$ at the $5\%$ significance level. A calculator may be used.

   (a)

   Assuming $\mathrm{H}_0$ is true, what is $\mathrm{P}(X\le10)$?

   [1 mark]

   • A$0.0171$
   • B$0.9520$
   • C$0.0480$
   • D$0.1133$

   (b)

   What is the critical region for the test?

   [1 mark]

   • A$X\le10$
   • B$X\le9$
   • C$X\le11$
   • D$X\ge11$

   (c)

   Exactly $10$ of the gardener's seeds germinate. State the conclusion of the test in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Emergency calls to a fire station arrive at random, with a mean of $2.5$ per hour. After a new housing estate opens, the controller thinks that the rate of calls has increased. In a randomly chosen $4$-hour period there are $16$ calls. Let $\lambda$ be the mean number of calls in a $4$-hour period and let $X$ be the number of calls in a $4$-hour period. The test is at the $5\%$ significance level. A calculator may be used.

   (a)

   Which pair of hypotheses should the controller use?

   [1 mark]

   • A$\mathrm{H}_0:\lambda=2.5,\ \mathrm{H}_1:\lambda>2.5$
   • B$\mathrm{H}_0:\lambda=10,\ \mathrm{H}_1:\lambda\ne10$
   • C$\mathrm{H}_0:\lambda=16,\ \mathrm{H}_1:\lambda>16$
   • D$\mathrm{H}_0:\lambda=10,\ \mathrm{H}_1:\lambda>10$

   (b)

   Assuming $\mathrm{H}_0$ is true, what is $\mathrm{P}(X\ge16)$?

   [1 mark]

   • A$0.0835$
   • B$0.0487$
   • C$0.9513$
   • D$0.9165$

   (c)

   Complete the test and state the conclusion in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A national survey reports that $40\%$ of Year 13 students study after 10 pm. A school believes that its own proportion is different. In a random sample of $150$ of its Year 13 students, $72$ say that they study after 10 pm. Let $p$ be the proportion of the school's Year 13 students who study after 10 pm and $X$ the number in the sample who do. The school tests $\mathrm{H}_0:p=0.4$ against $\mathrm{H}_1:p\ne0.4$ at the $5\%$ significance level, using a normal approximation to the binomial distribution. A calculator may be used.

   (a)

   Assuming $\mathrm{H}_0$ is true, use a normal approximation to find $\mathrm{P}(X\ge72)$.

   [3 marks]

   (b)

   Carry out the test and state the conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A call-centre manager states that $70\%$ of calls are answered within $20$ seconds. A customer group believes that the proportion is lower. It takes a random sample of $80$ calls and counts the number $X$ that are answered within $20$ seconds. Let $p$ be the proportion of all calls answered within $20$ seconds. The group tests the manager's claim at the $5\%$ significance level, using a normal approximation to the binomial distribution. A calculator may be used.

   (a)

   (i) Write down suitable hypotheses for the test.
   (ii) Use a normal approximation to find the critical region for the test.

   [6 marks]

   (b)

   In the sample, $46$ of the $80$ calls were answered within $20$ seconds.
   (i) Use the critical region to state the conclusion in context.
   (ii) Find the actual significance level of the test, using the normal approximation.
   (iii) Explain why the normal approximation is reasonable here.

   [6 marks]

   Total for question 4: 12 marks

End of questions