7 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.
- Find the probability that
(A) exactly one of the rocks contains fossils,
(B) at least one of the rocks contains fossils. - 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 \(n\).
- 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.