A dog breeder claims that the mean weight of male Great Dane dogs is $$\(20 \mathrm{~kg}\)$$ more than the mean weight of female Great Dane dogs. Tammy believes that the mean weight of male Great Dane dogs is more than $$\(20 \mathrm{~kg}\)$$ more than the mean weight of female Great Dane dogs. She takes random samples of 50 male and 50 female Great Dane dogs and records their weights. The results are summarised below, where $$\(x\)$$ denotes the weight, in $$\(\mathrm{kg}\)$$, of a male Great Dane dog and $$\(y\)$$ denotes the weight, in $$\(\mathrm{kg}\)$$, of a female Great Dane dog. $$\[ \sum x=3610 \quad \sum x^{2}=260955.6 \quad \sum y=2585 \quad \sum y^{2}=133757.2 \]$$ Stating your hypotheses clearly, carry out a suitable test to assess Tammy's belief. Use a $$\(5 \%\)$$ level of significance and state your critical value. (7)

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