题目
A dog breeder claims that the mean weight of male Great Dane dogs is 20 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 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 kg, of a male Great Dane dog and y denotes the weight, in kg, of a female Great Dane dog.
∑x = 3610 ∑x² = 260 955.6 ∑y = 2585 ∑y² = 133 757.2
(a) Find unbiased estimates for the mean and variance of the weights of
(i) the male Great Dane dogs,
(ii) the female Great Dane dogs.
(b) 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.
(c) For the test in part (b), state whether or not it is necessary to assume that the weights of the Great Dane dogs are normally distributed. Give a reason for your answer.
(d) State an assumption you have made in carrying out the test in part (b).
题目中文翻译
一名狗繁育者声称，雄性大丹犬的平均体重比雌性大丹犬的平均体重多 20 kg。
Tammy 认为，雄性大丹犬的平均体重比雌性大丹犬的平均体重多于 20 kg。
她随机抽取了 50 只雄性大丹犬和 50 只雌性大丹犬，并记录它们的体重。
结果汇总如下，其中 x 表示雄性大丹犬的体重（单位：kg），y 表示雌性大丹犬的体重（单位：kg）。
∑x = 3610 ∑x² = 260 955.6 ∑y = 2585 ∑y² = 133 757.2
(a) 求以下体重的均值和方差的无偏估计：
(i) 雄性大丹犬，
(ii) 雌性大丹犬。
(b) 清楚写出假设，进行一个适当的检验来评估 Tammy 的看法。使用 5% 显著性水平，并写出临界值。
(c) 对于 (b) 中的检验，是否有必要假设大丹犬的体重服从正态分布？说明理由。
(d) 写出你在进行 (b) 中检验时所作的一个假设。