Problem

A company manufactures copper pipes. The pipes are produced by two different machines, $A$ and $B$. An inspector claims that the mean diameter of the pipes produced by machine $A$ is greater than the mean diameter of the pipes produced by machine $B$. He takes a random sample of 12 pipes produced by machine $A$ and measures their diameters, $x$ cm. His results are summarised as follows.

$$\sum x = 6.24 \qquad \sum x^2 = 3.26$$

He also takes a random sample of 10 pipes produced by machine $B$ and measures their diameters in cm. His results are as follows.

$$\begin{gathered} 0.48,\ 0.53,\ 0.47,\ 0.54,\ 0.54,\\ 0.55,\ 0.46,\ 0.55,\ 0.50,\ 0.48. \end{gathered}$$

The diameters of the pipes produced by each machine are assumed to be normally distributed with equal population variances.

Test at the 2.5% significance level whether the data supports the inspector’s claim.