Seva is investigating the lengths of the tails of adult wallabies in two regions of Australia, \(X\) and \(Y\). He chooses a random sample of 50 adult wallabies from region \(X\) and records the lengths, \(x \mathrm{~cm}\), of their tails. He also chooses a random sample of 40 adult wallabies from region \(Y\) and records the lengths, \(y \mathrm{~cm}\), of their tails. His results are summarised as follows.
\(\sum x=1080 \quad \sum x^{2}=23480 \quad \sum y=940 \quad \sum y^{2}=22220\)

It cannot be assumed that the population variances of the two distributions are the same.
(a) Find a \(90 \%\) confidence interval for the difference between the population mean lengths of the tails of adult wallabies in regions \(X\) and \(Y\).

The population mean lengths of the tails of adult wallabies in regions \(X\) and \(Y\) are \(\mu_{X} \mathrm{~cm}\) and \(\mu_{Y} \mathrm{~cm}\) respectively.
(b) Test, at the \(10 \%\) significance level, the null hypothesis \(\mu_{Y}-\mu_{X}=1.1\) against the alternative hypothesis \(\mu_{Y}-\mu_{X}\gt 1.1\). State your conclusion in the context of the question.