In an ecological study of a specific butterfly population, it is found that 35% of the individuals carry a specific genetic marker, denoted by WWW. A researcher captures a random sample of 30 butterflies. Let the random variable XXX represent the number of butterflies in the sample who carry marker WWW.
Find (i) P(X=12)P(X = 12)P(X=12) (ii) P(X>8)P(X > 8)P(X>8) (iii) the smallest value of kkk such that P(X≤k)≥0.75P(X \le k) \ge 0.75P(X≤k)≥0.75
The study expands to a much larger region, and a new random sample of 400 butterflies is collected.
(i) Using a suitable normal approximation, find the probability that no more than 130 of these 400 butterflies carry marker WWW. (ii) Justify why a normal approximation is appropriate in part (b)(i).
