Dr. Aris is a marine biologist monitoring the health of coral colonies in a reef. He categorises colonies into four types based on their condition.
The distribution of these conditions among colonies in the reef is given in the table below.
| Condition | Healthy | Bleached | Algae-covered | Physically damaged |
|---|---|---|---|---|
| Probability | 0.32 | 0.44 | 0.14 | 0.10 |
Dr. Aris collects data from a random sample of 150 colonies.
Using a normal approximation, find the probability that in this sample:
(i) there are exactly 45 healthy colonies.
(ii) there are at most 25 colonies that are algae-covered.
(iii) there are at least 80 colonies which are not bleached.
Dr. Aris wants to investigate if conservation efforts have successfully reduced the proportion, ppp, of bleached colonies. He suspects that the proportion has decreased from its initial value of 0.44.
He formulates the following hypotheses:
H0:p=0.44 H_0: p = 0.44 H0:p=0.44 H1:p<0.44 H_1: p < 0.44 H1:p<0.44He takes a new large random sample of 300 colonies and finds that x x\,x of them are bleached.
(i) Using a 5% level of significance and a normal approximation, find the critical region for xxx.
(ii) In his sample, he finds that 115 colonies are bleached. Complete the test, stating the conclusion in context.
