An entomologist monitoring a woodland area observes that 15% of the local Emperor butterfly population exhibits a rare blue-tinged wing pattern. In a random survey of 30 butterflies, the random variable B B\,B represents the number of butterflies possessing this blue pattern.
Suggest a suitable distribution to model BBB.
State one condition necessary for the model in part (a) to be valid.
Calculate the probability that (i) exactly 27 butterflies in the survey possess the typical (non-blue) wing pattern, (ii) at least 7 butterflies possess the blue wing pattern.
A much larger study is conducted involving a random sample of 200 butterflies. The probability that at least k k\,k of these butterflies possess the blue wing pattern is greater than 0.8.
Using a suitable approximation, determine the maximum possible value of kkk.
The entomologist investigates a different forest site and suspects the frequency of the blue pattern has increased. In a random sample of 25 butterflies from this site, 8 are found to have the blue pattern.
Perform a statistical test at the 5% significance level to determine whether there is evidence that the proportion of butterflies with the blue pattern has increased. State your hypotheses clearly.
