The weight, \(X\) grams, of soup in a carton may be modelled by a normal random variable with mean 406 and standard deviation 4.2.
Find the probability that the weight of soup in a carton:
is less than 400 grams;
is between 402.5 grams and 407.5 grams.
The weight, \(Y\) grams, of chopped tomatoes in a tin is a normal random variable with mean \(\mu\) and standard deviation \(\sigma\).
Given that \(\mathrm { P } ( Y < 310 ) = 0.975\), explain why:
$$310 - \mu = 1.96 \sigma$$
Given that \(\mathrm { P } ( Y < 307.5 ) = 0.86\), find, to two decimal places, values for \(\mu\) and \(\sigma\).
(4 marks)