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)
7 (a) 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:\\
(i) is less than 400 grams;\\
(ii) is between 402.5 grams and 407.5 grams.\\
(b) The weight, $Y$ grams, of chopped tomatoes in a tin is a normal random variable with mean $\mu$ and standard deviation $\sigma$.\\
(i) Given that $\mathrm { P } ( Y < 310 ) = 0.975$, explain why:
$$310 - \mu = 1.96 \sigma$$
(ii) Given that $\mathrm { P } ( Y < 307.5 ) = 0.86$, find, to two decimal places, values for $\mu$ and $\sigma$.\\
(4 marks)
\hfill \mbox{\textit{AQA S1 Q7}}