7 It is known that on average \(85 \%\) of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.
- (A) Find the probability that exactly 12 germinate.
(B) Find the probability that fewer than 12 germinate.
The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting \(n\) of these seeds and sowing them. He then carries out a hypothesis test at the \(1 \%\) significance level to investigate whether he is correct. - Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
- In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test.
- Suppose instead that Ramesh conducts the trial with \(n = 50\), and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35 , complete the test.
- If \(n\) is small, there is no point in carrying out the test at the \(1 \%\) significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of \(n\) for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer.