A company produces bags of sugar with a labelled weight of 1 kg. The weights of the bags arenormally distributed with a mean of 1 kg and a standard deviation of 100 g. In an inspection, ifthe weight of a randomly chosen bag is less than 950 g then the company fails the inspection.

let $X$ be the weight of sugar in the bag

$\text{P}\left(X<950\right)=0.308537\dots \approx 0.309$ (M1)A1

[2 marks]

METHOD 1

let $\overline{X}$be the mean weight of $5$ bags of sugar

$\text{E}\left(\overline{X}\right)=1000$ (A1)

use of$\text{Var}\left(\overline{X}\right)=\frac{{\sigma }^2}{n}$ (M1)

$\text{Var}\left(\overline{X}\right)=\frac{{100}^2}{5}\left(=2000\right)$ (A1)

$\overline{X}~\text{N}\left(1000,2000\right)$

$\text{P}\left(\overline{X}>950\right)=0.868223\dots \approx 0.868\left(86.8\%\right)$ A1

METHOD 2

let $T$ be the total weight of $5$ bags of sugar

$\text{E}\left(T\right)=5000$ (A1)

use of$\text{Var}\left(X_1+X_2\right)=\text{Var}\left(X_1\right)+\text{Var}\left(X_2\right)$for independent random variables (M1)

$\text{Var}\left(T\right)=5\times {100}^2\left(=50 000\right)$ (A1)

$T~\text{N}\left(5000,50 000\right)$

$\text{P}\left(T>4750\right)=0.868223\dots \approx 0.868\left(86.8\%\right)$ A1

[4 marks]