题目

Problem

It is known that $1.8\%$ of children in a certain country have not been vaccinated against measles. A random sample of $200$ children in this country is chosen.

(a) Use a suitable approximating distribution to find the probability that there are fewer than $3$ children in the sample who have not been vaccinated against measles.

(4)

(b) Justify your approximating distribution.

(2)

题目中文翻译

已知某国有 $1.8\%$ 的儿童未接种麻疹疫苗。从该国随机抽取 $200$ 名儿童。

(a) 用适当的近似分布求样本中未接种麻疹疫苗的儿童少于 $3$ 人的概率。

(4)

(b) 说明采用该近似分布的理由。

(2)

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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未接种人数服从二项分布，大样本低比例用参数3.6的泊松近似，少于3即0至2。

答题过程

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Let $U$ be the number of unvaccinated children in the sample. Then $U\sim\mathrm{B}(200,0.018)$ . Use

U\approx\mathrm{Po}(200(0.018))=\mathrm{Po}(3.6).

Hence

\begin{align*} \mathrm{P}(U<3)\approx&\,e^{-3.6}\bigg(1+3.6+\frac{3.6^2}{2!}\bigg)\\[4mm] =&\,\boxed{0.303}\quad\text{(3 s.f.).} \end{align*}

(b)

解法一

思路

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用具体参数说明大n、小np的泊松近似条件。

答题过程

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The sample size is large, $n=200>50$ , and $np=200(0.018)=3.6<5$ , so a Poisson approximation is suitable.

CIE 9709 2022 June Paper 62 Q3

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程