For ten steel ingots from a production process the following measures of hardness were obtained:

$$73.2, \quad 74.3, \quad 75.4, \quad 73.8, \quad 74.4, \quad 76.7, \quad 76.1, \quad 73.0, \quad 74.6, \quad 74.1 .$$

On the assumption that the variation is well described by a normal density function obtain an estimate of the process mean.

The manufacturer claims that he is supplying steel with mean hardness 75 . Derive carefully a (generalized) likelihood ratio test of this claim. Knowing that for the data above

$$S_{X X}=\sum_{j=1}^{n}\left(X_{i}-\bar{X}\right)^{2}=12.824$$

what is the result of the test at the $5 \%$ significance level?

$\left.\begin{array}{lll}{[\text { Distribution }} & t_{9} & t_{10} \\ 95 \% \text { percentile } & 1.83 & 1.81 \\ 97.5 \% \text { percentile } & 2.26 & 2.23\end{array}\right]$