Hypothesis Testing - Assumptions and Validity of Mean Tests

A distributor for health products claimed that his slimming pills could help women achieve their desired mass of \(49\) kg. An experiment was carried out to test his claim. \(80\) women were randomly chosen and they took the slimming pills for a period of time. The mass, \(x\), of each woman was then taken and recorded in kg. it was calculated that \(\sum{(x-40)=837}\).

1. State, with a reason, whether it is necessary to assume that the masses of the women follows a normal distribution in order to test his claim at \(2\%\) significance level.
2. Assuming that the population variance of the masses is \(100\) kg\(^{2}\), test at \(4\%\) significance level whether the distributor was understating his claim.