题目
Duck eggs are classed as large or small.
Kiyoshi knows that 35% of the eggs laid by his ducks are classed as large and the rest are classed as small. The eggs are randomly allocated to boxes each containing 6 eggs.
A box is selected at random.
Calculate the probability that this box contains
(a) (i) exactly 2 large eggs,
(ii) more large eggs than small eggs.
Kiyoshi believes that the proportion of large eggs produced by the ducks will increase if he adds a supplement to the ducks’ food. To check his belief he takes a random sample of 50 eggs after the supplement has been added to the ducks’ food and finds that 25 of them are large eggs.
(b) Use a suitable test, at the 5% level of significance, to determine whether or not there is evidence to support Kiyoshi’s belief. State your hypotheses clearly.
Kiyoshi sells boxes that contain more large eggs than small eggs for a profit of £1.20 and the rest of the boxes for a profit of £0.60
The proportion of large eggs produced by Kiyoshi’s ducks has increased to 45% and the cost of the supplement is £0.10 per box of eggs. Given that the eggs are still randomly allocated to boxes,
(c) explain whether or not Kiyoshi should continue to add the supplement to the ducks’ food in order to make a greater profit.
(Total for Question 2 is 15 marks)
题目中文翻译
鸭蛋分为大蛋或小蛋。
Kiyoshi 知道，他的鸭子产的蛋中有 35% 是大蛋，其余是小蛋。鸡蛋被随机分配到每盒 6 个鸡蛋的盒子中。
随机选取一个盒子。
求这个盒子中包含以下情况的概率：
(a) (i) 恰好 2 个大蛋；
(ii) 大蛋比小蛋更多。
Kiyoshi 相信，如果给鸭子饲料添加一种补充剂，大蛋比例会增加。为了检验这一想法，他在添加补充剂后随机抽取 50 个蛋，其中 25 个是大蛋。
(b) 使用合适的检验，在 5% 显著性水平下判断是否有证据支持 Kiyoshi 的想法，并清楚写出假设。
Kiyoshi 以每盒 £1.20 的利润出售“大蛋多于小蛋”的盒子，其余盒子每盒利润为 £0.60。
Kiyoshi 鸭子产大蛋的比例已增加到 45%，而补充剂成本为每盒蛋 £0.10。假设鸡蛋仍随机分配到盒子中，
(c) 说明 Kiyoshi 是否应继续添加补充剂以获得更高利润。
（第 2 题共 15 分）