A restaurant states that its hamburgers contain 20\% fat. Paul claims that the mean fat content of their hamburgers is less than 20\%. Paul takes a random sample of 50 hamburgers from the restaurant and finds that they contain a mean fat content of 19.5\% with a standard deviation of 1.5\%

You may assume that the fat content of hamburgers is normally distributed.

\begin{enumerate}[label=(\alph*)]
\item Find the 90\% confidence interval for the mean fat content of hamburgers from the restaurant.
(4)

\item State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers.
(2)

The restaurant changes the mean fat content of their hamburgers to $\mu$\% and adjusts the standard deviation to 2\%. Paul takes a sample of size $n$ from this new batch of hamburgers. He uses the sample mean $\bar{X}$ as an estimator of $\mu$.

\item Find the minimum value of $n$ such that $\mathrm{P}(|\bar{X} - \mu| < 0.5) \geq 0.9$
(5)
\end{enumerate}

\hfill \mbox{\textit{Edexcel S3 2016 Q7}}