Hypothesis tests for binomial and PoissonEdexcel International A Level Further Maths: Mind map

Exam questions on Hypothesis tests for binomial and Poisson

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.
   Exactly $10$ of the gardener's seeds germinate. State the conclusion of the test in context.2 marks
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.
   Complete the test and state the conclusion in context.2 marks
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.
   Assuming $\mathrm{H}_0$ is true, use a normal approximation to find $\mathrm{P}(X\ge72)$.3 marks