A marine biologist is analyzing the composition of microplastics found in sediment samples from a specific coastal region. She categorizes the particles into four types based on their morphology.
The distribution of these types among the collected particles is shown in the following table:
| Particle Type | Microbeads | Fragments | Fibers | Pellets |
|---|---|---|---|---|
| Probability | 0.18 | 0.42 | 0.30 | 0.10 |
Samples are processed in batches of 150 particles.
Using a suitable approximation, find the probability that in a single batch:
(i) there are exactly 30 microbeads.
(ii) there are at most 55 fibers.
(iii) there are at least 95 particles which are not fragments.
The biologist investigates a different coastal region where she suspects the proportion, ppp, of fragment-type microplastics has decreased from the usual 0.42.
She tests the hypotheses:
H0:p=0.42 H_0: p = 0.42 H0:p=0.42 H1:p<0.42 H_1: p < 0.42 H1:p<0.42She takes a random sample of 400 particles and finds that x x\,x of them are fragments.
(i) Using a 5% level of significance and a normal approximation, find the critical region for xxx.
(ii) In her sample, she finds 150 fragments. Complete the test, stating your conclusion in context.
