N2008 P2 Q6

In mineral water from a certain source, the mass of calcium, \(X\) mg, in a one-litre bottle is a normally distributed random variable with mean \(\mu\). Based on observations over a long period, it is known that \(\mu=78\). Following a period of extreme weather, 15 randomly chosen bottles of the water were analysed. The masses of calcium in the bottles are summarised by \(\sum x=1026.0\), \(\sum x^2=77265.90\).

Test, at the 5% significance level, whether the mean mass of calcium in a bottle has changed.[6]