Infer a normal standard deviation from symmetric tail areas

A Kranji farm sells chickens and ducks.

(a) Chicken mass has mean 2.5 kg and standard deviation 0.4 kg. If the probability that the mean of a large sample of $n$ chickens exceeds 2.45 kg is greater than 0.95, find the least $n$.3 marks

(b) Duck mass $X$ kg is normal with mean 2.7 and standard deviation $\sigma$. Ducks cost $4 per kg. (i) Given $P(X<2.5)=\frac{50}{107}P(X<2.9)$, show that $\sigma=0.424$ to 3s.f. 2 marks (ii) A housewife brings $ 50. Find the probability she has less than $27.50 after buying2 ducks. 3 marks (iii) For 4 randomly chosen ducks, find the probability the most expensive costs less than $ 11.2 marks