2015 TJC P2 Q7 [Modified]

The manufacturer of Mola snack biscuits claimed that the mean biscuit content in a packet is \(15\)g. A sample of \(80\) packets is taken and the weight, \(x\)g, of the content in each packet is measured. The results are summarised by

\(\sum{\left( x-15 \right)=53.6}\) and \(\sum{{{\left( x-15 \right)}^{2}}=795.7}\).

1. Calculate unbiased estimates of the population mean and variance.
2. Stating a necessary assumption, test whether the mean weight differs from \(15\)g at \(4%\) level of significance.
3. If the test is conducted at \(\alpha %\) level of significance and the null hypothesis is rejected in favour of the alternative hypothesis based on the values from this sample, determine the set of values of \(\alpha \).
4. Another large random sample of \(n\) packets is taken. Assume now that the population standard deviation is \(3\)g. Find the range of values of the sample mean, in terms of \(n\), for which the claim that the mean weight differs from \(15\)g is supported at \(4%\) level of significance.