Problem

A company has two machines, A and B, which independently fill small bottles with a liquid. The volumes of liquid per bottle, in suitable units, filled by machines A and B are denoted by $x$ and $y$ respectively. A scientist at the company takes a random sample of 40 bottles filled by machine A and a random sample of 50 bottles filled by machine B. The results are summarised as follows.

$\sum x = 1120 \qquad \sum x^2 = 31400 \qquad \sum y = 1370 \qquad \sum y^2 = 37600$

The population means of the volumes of liquid in the bottles filled by machines A and B are denoted by $\mu_A$ and $\mu_B$.

(a) Test at the 2% significance level whether there is any difference between $\mu_A$ and $\mu_B$.

(b) Find the set of values of $\alpha$ for which there would be evidence at the $\alpha\%$ significance level that $\mu_A - \mu_B$ is greater than 0.25.