5 A general store sells lawn fertiliser in 2.5 kg bags, 5 kg bags and 10 kg bags.
\begin{enumerate}[label=(\alph*)]
\item The actual weight, $W$ kilograms, of fertiliser in a 2.5 kg bag may be modelled by a normal random variable with mean 2.75 and standard deviation 0.15 .

Determine the probability that the weight of fertiliser in a 2.5 kg bag is:
\begin{enumerate}[label=(\roman*)]
\item less than 2.8 kg ;
\item more than 2.5 kg .
\end{enumerate}\item The actual weight, $X$ kilograms, of fertiliser in a 5 kg bag may be modelled by a normal random variable with mean 5.25 and standard deviation 0.20 .
\begin{enumerate}[label=(\roman*)]
\item Show that $\mathrm { P } ( 5.1 < X < 5.3 ) = 0.372$, correct to three decimal places.
\item A random sample of four 5 kg bags is selected. Calculate the probability that none of the four bags contains between 5.1 kg and 5.3 kg of fertiliser.
\end{enumerate}\item The actual weight, $Y$ kilograms, of fertiliser in a 10 kg bag may be modelled by a normal random variable with mean 10.75 and standard deviation 0.50.

A random sample of six 10 kg bags is selected. Calculate the probability that the mean weight of fertiliser in the six bags is less than 10.5 kg .
\end{enumerate}

\hfill \mbox{\textit{AQA S1 2012 Q5 [13]}}