题目

Problem

The weights, in kg, of bags of rice produced by Anders have the distribution $\mathrm{N}(2.02,\,0.03^2)$ .

(a) Find the probability that a randomly chosen bag of rice produced by Anders weighs between 1.98 and $2.03\,\mathrm{kg}$ .

(3)

The weights of bags of rice produced by Binders are normally distributed with mean $2.55\,\mathrm{kg}$ and standard deviation $\sigma\,\mathrm{kg}$ . In a random sample of 5000 of these bags, 134 weighed more than $2.6\,\mathrm{kg}$ .

(b) Find the value of $\sigma$ .

(4)

题目中文翻译

Anders 生产的袋装米的重量以千克为单位，服从分布 $\mathrm{N}(2.02,\,0.03^2)$ 。

(a) 求随机选取的一袋 Anders 生产的米的重量在 1.98 至 $2.03\,\mathrm{kg}$ 之间的概率。

(3)

Binders 生产的袋装米的重量服从正态分布，平均数为 $2.55\,\mathrm{kg}$ ，标准差为 $\sigma\,\mathrm{kg}$ 。在随机抽取的 5000 袋米的样本中，有 134 袋的重量超过 $2.6\,\mathrm{kg}$ 。

(b) 求 $\sigma$ 的值。

(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

展开

分别标准化两个边界，两者并不关于均值对称。

答题过程

展开

For $W\sim\mathrm{N}(2.02,0.03^2)$ ,

\begin{align*} \mathrm{P}(1.98<W<2.03)=&\,\Phi\biggl(\frac{2.03-2.02}{0.03}\biggr)\\[4mm] &\,\hspace{2pt}-\Phi\biggl(\frac{1.98-2.02}{0.03}\biggr)\\[4mm] =&\,\Phi(1/3)-\Phi(-4/3)\\[4mm] =&\,0.539347440\ldots\approx\boxed{0.539}. \end{align*}

(b)

解法一

思路

展开

先把134/5000右尾转换为左尾，再反查分位点求标准差。

答题过程

展开

Use the sample proportion to estimate

\mathrm{P}(W>2.6)=\frac{134}{5000}=0.0268.

Therefore

\frac{2.6-2.55}{\sigma}=\Phi^{-1}(0.9732)=1.930055188\ldots.

Hence

\sigma=\frac{0.05}{1.930055188\ldots} =0.025905995\ldots\approx\boxed{0.0259\text{ kg}}.

CIE 9709 2022 June Paper 52 Q4

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程