Find sample size for test

2 A die has six faces numbered \(1,2,3,4,5,6\). Manjit suspects that the die is biased so that it shows a six on fewer throws than it would if it were fair. In order to test her suspicion, she throws the die a certain number of times and counts the number of sixes.

1. State suitable null and alternative hypotheses for Manjit's test.
2. There are no sixes in the first 15 throws. Show that this result is not significant at the \(5 \%\) level.
3. Find the smallest value of \(n\) such that, if there are no sixes in the first \(n\) throws, this result is significant at the 5\% level.

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.

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.

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.

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.

1. (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.
2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
3. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test.
4. 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.
5. 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.

12 The jaguar is a species of big cat native to South America. Records show that 6\% of jaguars are born with black coats. Jaguars with black coats are known as black panthers. Due to deforestation a population of jaguars has become isolated in part of the Amazon basin. Researchers believe that the percentage of black panthers may not be \(6 \%\) in this population.

1. Find the minimum sample size needed to conduct a two-tailed test to determine whether there is any evidence at the \(5 \%\) level to suggest that the percentage of black panthers is not \(6 \%\). A research team identifies 70 possible sites for monitoring the jaguars remotely. 30 of these sites are randomly selected and cameras are installed. 83 different jaguars are filmed during the evidence gathering period. The team finds that 10 of the jaguars are black panthers.
2. Conduct a hypothesis test to determine whether the information gathered by the research team provides any evidence at the \(5 \%\) level to suggest that the percentage of black panthers in this population is not \(6 \%\).

1. Past evidence shows that \(7 \%\) of pears grown by a farmer are unfit for sale.

This season it is believed that the proportion of pears that are unfit for sale has decreased. To test this belief a random sample of \(n\) pears is taken. The random variable \(Y\) represents the number of pears in the sample that are unfit for sale.

1. Find the smallest value of \(n\) such that \(Y = 0\) lies in the critical region for this test at a \(5 \%\) level of significance. In the past, \(8 \%\) of the pears grown by the farmer weigh more than 180 g . This season the farmer believes the proportion of pears weighing more than 180 g has changed. She takes a random sample of 75 pears and finds that 11 of them weigh more than 180 g .
2. Test, using a suitable approximation, whether there is evidence of a change in the proportion of pears weighing more than 180 g .
   You should use a \(5 \%\) level of significance and state your hypotheses clearly.