6 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.

1. Find a \(90 \%\) confidence interval for the difference between the population mean lengths of the tails of adult wallabies in regions \(X\) and \(Y\).
   \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-10_2718_38_141_2010} 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.
2. Test, at the \(10 \%\) significance level, the null hypothesis \(\mu _ { Y } - \mu _ { X } = 1.1\) against the alternative hypothesis \(\mu _ { Y } - \mu _ { X } > 1.1\). State your conclusion in the context of the question.
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