OCR MEI S1 (Statistics 1)

1 A multinational accountancy firm receives a large number of job applications from graduates each year. On average \(20 \%\) of applicants are successful. A researcher in the human resources department of the firm selects a random sample of 17 graduate applicants.

1. Find the probability that at least 4 of the 17 applicants are successful.
2. Find the expected number of successful applicants in the sample.
3. Find the most likely number of successful applicants in the sample, justifying your answer. It is suggested that mathematics graduates are more likely to be successful than those from other fields. In order to test this suggestion, the researcher decides to select a new random sample of 17 mathematics graduate applicants. The researcher then carries out a hypothesis test at the \(5 \%\) significance level.
4. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Give a reason for your choice of the alternative hypothesis.
5. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
6. Explain why the critical region found in part (v) would be unaltered if a \(10 \%\) significance level were used.

2 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.

1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
   (A) both seeds germinate,
   (B) just one seed germinates,
   (C) neither seed germinates.
2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.

3 Douglas plays darts, and the probability that he hits the number he is aiming at is 0.87 for any particular dart. Douglas aims a set of three darts at the number 20 ; the number of times he is successful can be modelled by \(\mathrm { B } ( 3,0.87 )\).

1. Calculate the probability that Douglas hits 20 twice.
2. Douglas aims fifty sets of 3 darts at the number 20. Find the expected number of sets for which Douglas hits 20 twice.
3. Douglas aims four sets of 3 darts at the number 20. Calculate the probability that he hits 20 twice for two sets out of the four.

4 A geologist splits rocks to look for fossils. On average 10\% of the rocks selected from a particular area do in fact contain fossils. The geologist selects a random sample of 20 rocks from this area.

1. Find the probability that
   (A) exactly one of the rocks contains fossils,
   (B) at least one of the rocks contains fossils.
2. A random sample of \(n\) rocks is selected from this area. The geologist wants to have a probability of 0.8 or greater of finding fossils in at least one of the \(n\) rocks. Find the least possible value of \(n\).
3. The geologist explores a new area in which it is claimed that less than \(10 \%\) of rocks contain fossils. In order to investigate the claim, a random sample of 30 rocks from this area is selected, and the number which contain fossils is recorded. A hypothesis test is carried out at the 5\% level.
   (A) Write down suitable hypotheses for the test.
   (B) Show that the critical region consists only of the value 0 .
   (C) In fact, 2 of the 30 rocks in the sample contain fossils. Complete the test, stating your conclusions clearly.