题目

Problem

The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean $28\,\mathrm{kg}$ . On 10% of days less than $16\,\mathrm{kg}$ of grapes are sold.

(a) Find the standard deviation of the mass of grapes sold per day.

(3)

The mass of grapes sold on any day is independent of the mass sold on any other day.

(b) 12 days are chosen at random.

Find the probability that less than $16\,\mathrm{kg}$ of grapes are sold on more than 2 of these 12 days.

(3)

(c) In a random sample of 365 days, on how many days would you expect the mass of grapes sold to be within 1.3 standard deviations of the mean?

(4)

题目中文翻译

一家大型商店每天售出的葡萄质量可用均值为 $28\,\mathrm{kg}$ 的正态分布建模。在 10% 的日子里，售出的葡萄少于 $16\,\mathrm{kg}$ 。

(a) 求每天售出葡萄质量的标准差。

(3)

任意一天售出葡萄的质量与其他任何一天售出的质量相互独立。

(b) 随机选择 12 天。

求这 12 天中有超过 2 天售出的葡萄少于 $16\,\mathrm{kg}$ 的概率。

(3)

(4)

原卷通用作答要求

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

解答

(a)

解法一

思路

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16对应左尾0.10，分位点为负，所得标准差为正。

答题过程

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\frac{16-28}{\sigma}=\Phi^{-1}(0.10)=-1.281551565\ldots.

Thus

\sigma=\frac{-12}{-1.281551565\ldots} =9.363649752\ldots\approx\boxed{9.36\text{ kg}}.

(b)

解法一

思路

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一天低于16的概率直接用0.10，12天中超过2天用0、1、2天的补集。

答题过程

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Let $D\sim\mathrm{B}(12,0.1)$ . Then

\begin{align*} \mathrm{P}(D>2) =&\,1-\bigl[(0.9)^{12}+12(0.1)(0.9)^{11}\\[4mm] &\,\hspace{2pt}+\binom{12}{2}(0.1)^2(0.9)^{10}\bigr]\\[4mm] =&\,0.110869977\ldots\approx\boxed{0.111}. \end{align*}

(c)

解法一

思路

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距均值1.3个标准差内对应标准正态−1.3至1.3，概率不依赖(a)的标准差数值。

答题过程

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The required probability is

\begin{align*} p=&\,\mathrm{P}(-1.3<Z<1.3)\\[4mm] =&\,2\Phi(1.3)-1=0.806399030\ldots. \end{align*}

The expected number of days is

365p=294.335646\ldots.

Thus we would expect 294 days.

CIE 9709 2023 June Paper 53 Q6

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程