1 A factory produces bottles of brown sauce and bottles of tomato sauce.
\begin{enumerate}[label=(\alph*)]
\item The content, $Y$ grams, of a bottle of brown sauce is normally distributed with mean $\mu _ { Y }$ and variance 4.

A quality control inspection found that the mean content, $\bar { y }$ grams, of a random sample of 16 bottles of brown sauce was 450 .

Construct a $95 \%$ confidence interval for $\mu _ { Y }$.
\item The content, $X$ grams, of a bottle of tomato sauce is normally distributed with mean $\mu _ { X }$ and variance $\sigma ^ { 2 }$.

A quality control inspection found that the content, $x$ grams, of a random sample of 9 bottles of tomato sauce was summarised by

$$\sum x = 4950 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 334$$
\begin{enumerate}[label=(\roman*)]
\item Construct a 90\% confidence interval for $\mu _ { X }$.
\item Holly, the supervisor at the factory, claims that the mean content of a bottle of tomato sauce is 545 grams.

Comment, with a justification, on Holly's claim. State the level of significance on which your conclusion is based.\\
(3 marks)
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S2 2011 Q1 [11]}}