题目

Problem

The masses, in grams, of apples of a certain type have the distribution $\mathrm{N}(58.4,6.3^2)$ . The apples are packed in bags, with each bag containing $10$ randomly chosen apples.

Find the probability that a randomly chosen bag contains apples whose total mass is between $550\,\mathrm{g}$ and $600\,\mathrm{g}$ .

(5)

题目中文翻译

某种苹果的质量（以克为单位）服从分布 $\mathrm{N}(58.4,6.3^2)$ 。这些苹果被装入袋中，每袋装有 $10$ 个随机选取的苹果。

求随机选取的一袋苹果的总质量介于 $550\,\mathrm{g}$ 与 $600\,\mathrm{g}$ 之间的概率。

(5)

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

解法一

思路

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总质量是十个独立正态变量之和：均值相加，方差相加，标准差不能直接相加。这里质量连续，不做连续性修正。

答题过程

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Let $T$ be the total mass of the ten randomly chosen apples, treating their masses as independent. Then

\begin{align*} \mathrm E(T)=&\,10(58.4)=584,\\[4mm] \operatorname{Var}(T)=&\,10(6.3^2)=396.9. \end{align*}

A sum of independent normal variables is normal, so $T\sim\mathrm N(584,396.9)$ . Thus

\begin{align*} \mathrm P(550<T<600) =&\,\mathrm P\bigg(Z<\frac{600-584}{\sqrt{396.9}}\bigg)\\[4mm] &\,-\mathrm P\bigg(Z\le\frac{550-584}{\sqrt{396.9}}\bigg)\\[4mm] =&\,\Phi(0.8031)-\Phi(-1.7066)\\[4mm] =&\,\Phi(0.8031)+\Phi(1.7066)-1\\[4mm] \approx&\,\boxed{0.745}. \end{align*}

CIE 9709 2026 June Paper 61 Q1

题目

解答

解法一

思路

答题过程