4 A machine is set to produce metal discs with mean diameter 15.4 mm . In order to test the correctness of the setting, a random sample of 12 discs was selected and the diameters, \(x \mathrm {~mm}\), were measured. The results are summarised by \(\Sigma x = 177.6\) and \(\Sigma x ^ { 2 } = 2640.40\). Diameters may be assumed to be normally distributed with mean \(\mu \mathrm { mm }\).

1. Find a \(95 \%\) confidence interval for \(\mu\).
2. Test, at the \(5 \%\) significance level, the null hypothesis \(\mu = 15.4\) against the alternative hypothesis \(\mu < 15.4\).