Roastie's Coffee is sold in packets with a stated weight of 250 g . A supermarket manager claims that the mean weight of the packets is less than the stated weight. She weighs a random sample of 90 packets from their stock and finds that their weights have a mean of 248 g and a standard deviation of 5.4 g .
Using a \(5 \%\) level of significance, test whether or not the manager's claim is justified. State your hypotheses clearly.
Find the \(98 \%\) confidence interval for the mean weight of a packet of coffee in the supermarket's stock.
State, with a reason, the action you would recommend the manager to take over the weight of a packet of Roastie's Coffee.
Roastie's Coffee company increase the mean weight of their packets to \(\mu \mathrm { g }\) and reduce the standard deviation to 3 g . The manager takes a sample of size \(n\) from these new packets. She uses the sample mean \(\bar { X }\) as an estimator of \(\mu\).
Find the minimum value of \(n\) such that \(\mathrm { P } ( | \bar { X } - \mu ! | < 1 ) \geqslant 0.98\)