Recover normal parameters and combine independent masses

(i) Factory A’s nut mass is normal with mean $\mu$ grams and standard deviation $\sigma$ grams. For a random sample of 50, $P(\bar X<247)=0.018079$ and $P(\sum X_j>12600)=0.78397$, each correct to 5 s.f. Find $\mu,\sigma$ to the nearest gram.5 marks

(ii) State clearly the parameters of every normal distribution used. Factory B’s independent mass models are:

Product	Mean (g)	Standard deviation (g)
Bolt	745	7.3
Nut	250	5

(a) Find the probability that a randomly chosen bolt’s mass differs from three times one randomly chosen nut’s mass by at least 40 g.4 marks

(b) A new process reduces each nut’s mass by 10%. Find the probability that the total mass of 10 randomly chosen new nuts is below 2.24 kg.3 marks