题目

Problem

In a certain country, the heights of the adult population are normally distributed with mean $1.64\,\mathrm{m}$ and standard deviation $0.25\,\mathrm{m}$ .

(a) Find the probability that an adult chosen at random from this country will have height greater than $1.93\,\mathrm{m}$ .

(3)

In another country, the heights of the adult population are also normally distributed. $33\%$ of the adult population have height less than $1.56\,\mathrm{m}$ . $25\%$ of the adult population have height greater than $1.86\,\mathrm{m}$ .

(b) Find the mean and the standard deviation of this distribution.

(5)

题目中文翻译

在某个国家，成年人口的身高服从均值为 $1.64\,\mathrm{m}$ 、标准差为 $0.25\,\mathrm{m}$ 的正态分布。

(a) 求从该国随机选出的一名成年人身高大于 $1.93\,\mathrm{m}$ 的概率。

(3)

在另一个国家，成年人口的身高也服从正态分布。该国 $33\%$ 的成年人身高小于 $1.56\,\mathrm{m}$ ， $25\%$ 的成年人身高大于 $1.86\,\mathrm{m}$ 。

(b) 求这个分布的均值和标准差。

(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

(a)

解法一

思路

展开

标准化1.93，取右尾概率。

答题过程

展开

Let $H\sim\mathrm{N}(1.64,0.25^2)$ . Then

\begin{align*} \mathrm{P}(H>1.93) =&\,1-\Phi\biggl(\frac{1.93-1.64}{0.25}\biggr)\\[4mm] =&\,1-\Phi(1.16)\\[4mm] =&\,0.123024\ldots\\[4mm] \approx&\,\boxed{0.123}. \end{align*}

(b)

解法一

思路

展开

第二个条件是右尾0.25，换成左尾0.75；用两个分位点列联立方程。

答题过程

展开

For the second country, write $H\sim\mathrm{N}(\mu,\sigma^2)$ . The given probabilities give

\frac{1.56-\mu}{\sigma}=\Phi^{-1}(0.33)=-0.439913\ldots,

\frac{1.86-\mu}{\sigma}=\Phi^{-1}(0.75)=0.674489\ldots.

Subtracting,

\sigma=\frac{1.86-1.56}{0.674489\ldots+0.439913\ldots} =0.269202454\ldots.

Then

\mu=1.56+0.439913\ldots(0.269202454\ldots) =1.678425703\ldots.

Thus the mean is $1.68\,\mathrm{m}$ and the standard deviation is $0.269\,\mathrm{m}$ .

CIE 9709 2024 June Paper 53 Q2

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程