Problem

Dev owns a small company which produces bottles of juice. He uses two machines, $X$ and $Y$, to fill empty bottles with juice. Dev is investigating the volumes of juice in the bottles. He chooses a random sample of 35 bottles filled by machine $X$ and a random sample of 60 bottles filled by machine $Y$. The volumes of juice, $x$ and $y$ respectively, measured in suitable units, are summarised by

$$\begin{gathered} \sum x = 30.8, \qquad \sum x^2 = 29.0, \\ \sum y = 62.4, \qquad \sum y^2 = 76.8. \end{gathered}$$

Dev claims that the mean volume of juice in bottles filled by machine $Y$ is greater than the mean volume of juice in bottles filled by machine $X$. A test at the $\alpha$% significance level suggests that there is sufficient evidence to support Dev’s claim.

Find the set of possible values of $\alpha$.