题目

Problem

$X$ is a random variable with distribution $\mathrm{B}(10,0.2)$ . A random sample of $160$ values of $X$ is taken.

(a) Find the approximate distribution of the sample mean, including the values of the parameters.

(3)

(b) Hence find the probability that the sample mean is less than $1.8$ .

(3)

题目中文翻译

随机变量 $X$ 服从 $\mathrm{B}(10,0.2)$ 分布。从 $X$ 中随机抽取 $160$ 个取值作为样本。

(a) 求样本均值的近似分布，并给出参数值。

(3)

(b) 由此求样本均值小于 $1.8$ 的概率。

(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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单个二项变量均值2、方差1.6，中心极限定理给出160个值的均值近似正态，方差除以160。

答题过程

展开

For one value,

\begin{align*} \mathrm{E}(X)=&\,10(0.2)=2,\\[4mm] \mathrm{Var}(X)=&\,10(0.2)(0.8)=1.6. \end{align*}

By the Central Limit Theorem, the sample mean has approximate distribution

\begin{align*} \overline X\approx&\,\mathrm{N}\bigg(2,\frac{1.6}{160}\bigg)\\[4mm] =&\,\boxed{\mathrm{N}(2,0.01)}. \end{align*}

(b)

解法一

思路

展开

样本总和为整数，均值严格小于1.8等价于总和不超过287。将边界287.5换回均值尺度再标准化。

答题过程

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Let $S=160\overline X$ be the sample total. Since $S$ is an integer,

\begin{align*} \overline X<1.8\quad\Longleftrightarrow&\,S<288\\[4mm] \Longleftrightarrow&\,S\leq287. \end{align*}

Using the normal approximation from part (a) with a continuity correction,

\begin{align*} &\,\mathrm{P}(\overline X<1.8)\\[4mm] \approx&\,\Phi\bigg(\frac{287.5/160-2}{0.1}\bigg)\\[4mm] =&\,\Phi(-2.03125)\\[4mm] =&\,0.0211148\ldots\\[4mm] =&\,\boxed{0.0211}\quad\text{(3 s.f.).} \end{align*}

CIE 9709 2022 November Paper 62 Q5

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程