2024 TJC H1 Q11

In a factory, salt is packed into bags of \(25\) grams by a filling machine.

1. The quality control manager selects a random sample of \(50\) bags of salt. The masses, \(x\) grams, are summarized by\[\sum x = 995.7 \text{ and}\hspace{0.5em} \sum x^2 = 26884.37\]Test, at the \(5\%\) level of significance, whether the filling machine produces underweight bags of salt.[7]
2. Following a slight adjustment to the filling machine and assuming that the mass of salt in a bag follows a normal distribution, the quality control manager takes a new random sample of \(50\) bags to check if the mean mass of salt per bag is \(25\) grams. The mean and standard deviation of the sample is found to be \(22\) grams and \(12.0\) grams respectively. A test at the \(\alpha\%\) significance level gives insufficient evidence to conclude that the mean mass of salt in each bag differs from \(25\) grams. Find the range of possible values of \(\alpha\).[4]