题目

Problem

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.

(5)

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.

(5)

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.

(5)

(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。鸭蛋被随机分配到每盒 6 个蛋的盒子中。

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。假设鸭蛋仍随机分配到盒子中，

（第 2 题共 15 分）

解答

(a)(i)

解法一

思路

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每个蛋是大蛋的概率固定为 0.35，且每盒随机装入 6 个蛋，因此一盒中的大蛋数量服从二项分布。使用二项分布概率公式计算恰好 2 个大蛋的概率。

答题过程

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Let $E$ be the number of large eggs in a box. Then

E\sim\operatorname{B}(6,0.35).

Therefore,

\begin{align*} P(E=2) =&\,\binom62(0.35)^2(0.65)^4 \\[4mm] =&\,0.328005\ldots \\[4mm] =&\,\boxed{0.328}\quad\text{(3 s.f.)}. \end{align*}

(a)(ii)

解法一

思路

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一盒共有 6 个蛋，大蛋比小蛋更多意味着大蛋数量至少为 4。用补集 $1-P(E\leqslant3)$ 可避免分别计算三项。

答题过程

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More large eggs than small eggs means $E\geqslant4$ . Hence

\begin{align*} P(E\geqslant4) =&\,1-P(E\leqslant3) \\[4mm] =&\,1-0.882576\ldots \\[4mm] =&\,0.117423\ldots \\[4mm] =&\,\boxed{0.117}\quad\text{(3 s.f.)}. \end{align*}

(b)

解法一

思路

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“大蛋比例会增加”对应上尾单侧检验。以添加补充剂后抽到的大蛋数量作为检验统计量，在原假设下使用二项分布计算观测到 25 个或更多大蛋的概率，再与 5% 比较。

答题过程

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Let $p$ be the proportion of large eggs after the supplement is added. The hypotheses are

H_0:p=0.35, \qquad H_1:p>0.35.

Under $H_0$ , let

L\sim\operatorname{B}(50,0.35).

The p-value is

\begin{align*} P(L\geqslant25) =&\,1-P(L\leqslant24) \\[4mm] =&\,1-0.9793 \\[4mm] =&\,0.0207. \end{align*}

Since $0.0207<0.05$ , reject $H_0$ . There is sufficient evidence at the 5% significance level to support Kiyoshi’s belief that the proportion of large eggs increases after adding the supplement.

解法二

思路

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也可以按官方评分资料给出的临界域路线，寻找使上尾概率不超过 0.05 的最小整数。检查相邻值可确定临界域，再判断观测值 25 是否落入其中。

答题过程

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Use the same hypotheses, with $L\sim\operatorname{B}(50,0.35)$ under $H_0$ . Now,

P(L\geqslant24)=0.0396<0.05,

whereas

P(L\geqslant23)=0.0710>0.05.

Therefore, the critical region is

L\geqslant24.

Since the observed value $25$ lies in the critical region, reject $H_0$ . There is sufficient evidence that the proportion of large eggs has increased after adding the supplement.

(c)

解法一

思路

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先分别求添加补充剂前后“一盒中大蛋多于小蛋”的概率，再计算由概率提高带来的每盒预期毛利润增量。若增量小于每盒补充剂成本 £0.10，就不应继续添加。

答题过程

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Before adding the supplement, from part (a)(ii),

P(\text{more large eggs than small eggs})=0.117423\ldots

The expected profit per box before the supplement is

\begin{align*} 0.117423(1.20) +(1-0.117423)(0.60) =&\,\pounds0.67045\ldots \end{align*}

After adding the supplement, let $X\sim\operatorname{B}(6,0.45)$ . Then

P(X\geqslant4)=0.255263\ldots

The expected gross profit per box is therefore

\begin{align*} 0.255263(1.20) +(1-0.255263)(0.60) =&\,\pounds0.75316\ldots \end{align*}

The increase in expected gross profit is

\pounds0.75316-\pounds0.67045 =\pounds0.08271.

Since $\pounds0.08271<\pounds0.10$ , the increased expected profit does not cover the supplement cost. Therefore, Kiyoshi should not continue to add the supplement.

解法二

思路

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也可以直接从添加补充剂后的预期毛利润中扣除 £0.10，得到每盒预期净利润，再与添加前的预期利润比较。

答题过程

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Using $P(X\geqslant4)=0.255263\ldots$ , the expected net profit per box after adding the supplement is

\begin{align*} 0.255263(1.20) +(1-0.255263)(0.60)-0.10 =&\,\pounds0.65316\ldots \end{align*}

This is less than the expected profit before the supplement,

\pounds0.65316<\pounds0.67045.

Therefore, Kiyoshi should not continue to add the supplement.

IAL 2020 Jan S2 Q2

题目

解答

(a)(i)

解法一

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(a)(ii)

解法一

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(b)

解法一

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解法二

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(c)

解法一

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解法二

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