Solve a normal interval condition and assess the model

In the problem’s model, birth weights for babies of non-smoking mothers at a hospital are normal with mean 3.05 kg and variance $\sigma^2$. The hospital’s stated classifications are:

Birth weight	Classification in the problem
Below 1.5 kg	Very low
1.5 kg to 2.5 kg	Low
2.5 kg to 4.0 kg	Normal
Above 4.0 kg	High

(i) A sample suggests 20.2% are in the low category. Assuming this holds for the population, find two possible $\sigma$ values to 2 dp and explain why one should be rejected.4 marks

Babies of smoking mothers have mean 2.80 kg. For the remaining parts, explicitly assume both populations are normal with standard deviation 0.86 kg.

(ii) Three smoking mothers are chosen at random. Find the probability all their babies are outside the normal-weight category, stating a required assumption.3 marks

(iii) Find the probability that the average weight of two babies of non-smoking mothers differs from twice the weight of one baby of a smoking mother by less than 2 kg.3 marks

Babies outside the normal category remain in hospital for observation until stable. Depending on treatment and length of stay, hospitalisation cost per baby has mean $2800 and standard deviation $ 500.

(iv) For 50 such babies, find the probability that average cost exceeds $3000.2 marks