Suppose that 30% of the population carry the gene for a certain disease. We consider a random sample of 20 individuals.

(a) What is the distribution of the number of people that carry the gene for the disease?

(b) In the observed sample, 8 subjects carry the gene. What is the estimated sample proportion of gene carriers?

(c) What is the exact probability that the sample proportion of gene carriers is bigger than 25%?

(d) Consider the distribution of the sampling proportion of gene carriers. Is it appropriate to use the normal approximation for its distribution? If yes, compute the approximate probability that the proportion of gene carriers is bigger than 25%. If no, justify your answer.