7. 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.

1. Find the \(90 \%\) confidence interval for the mean fat content of hamburgers from the restaurant.
2. State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers. 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 \(\mu\).
3. Find the minimum value of \(n\) such that \(\mathrm { P } ( | \bar { X } - \mu | < 0.5 ) \geqslant 0.9\)