题目

Problem

The heights, in metres, of white pine trees are normally distributed with mean $19.8$ and standard deviation $2.4$ .

In a certain forest there are 450 white pine trees.

(a) How many of these trees would you expect to have height less than $18.2$ metres?

(4)

The heights, in metres, of red pine trees are normally distributed with mean $23.4$ and standard deviation $\sigma$ . It is known that $26\%$ of red pine trees have height greater than $25.5$ metres.

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

(3)

题目中文翻译

白松树的高度（以米为单位）服从均值为 $19.8$ 、标准差为 $2.4$ 的正态分布。

某片森林中有 450 棵白松树。

(a) 预计其中有多少棵树的高度小于 $18.2$ 米？

(4)

红松树的高度（以米为单位）服从均值为 $23.4$ 、标准差为 $\sigma$ 的正态分布。已知 $26\%$ 的红松树高度大于 $25.5$ 米。

(b) 求 $\sigma$ 的值。

(3)

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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先求单棵树低于边界的概率，保留完整精度再乘450，最终给出整数棵数。

答题过程

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Let $H\sim\mathrm{N}(19.8,2.4^2)$ . Then

\begin{align*} \mathrm{P}(H<18.2) =&\,\Phi\biggl(\frac{18.2-19.8}{2.4}\biggr)\\[4mm] =&\,\Phi(-0.666666\ldots)=0.252492\ldots. \end{align*}

The expected number is

450(0.252492\ldots)=113.621641\ldots.

Thus we would expect 114 trees.

(b)

解法一

思路

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右尾0.26对应左尾0.74，用正分位点求正标准差。

答题过程

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For the height $R$ of a red pine tree,

\mathrm{P}(R\leq25.5)=0.74.

Therefore

\frac{25.5-23.4}{\sigma}=\Phi^{-1}(0.74)=0.643345\ldots,

\sigma=\frac{2.1}{0.643345\ldots} =3.264187\ldots\approx\boxed{3.26}.

CIE 9709 2024 November 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