题目

Problem

$1.6\%$ of adults in a certain town ride a bicycle. A random sample of $200$ adults from this town is selected.

(a) Use a suitable approximating distribution to find the probability that more than $3$ of these adults ride a bicycle.

(4)

(b) Justify your approximating distribution.

(2)

题目中文翻译

某镇有 $1.6\%$ 的成年人骑自行车。从这个镇随机抽取 $200$ 名成年人。

(a) 用适当的近似分布求这 $200$ 人中骑自行车的超过 $3$ 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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二项大样本小比例用参数3.2的泊松分布近似，超过3取0至3的补集。

答题过程

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Let $C$ be the number of cyclists in the sample. Then $C\sim\mathrm{B}(200,0.016)$ . Use

C\approx\mathrm{Po}(200(0.016))=\mathrm{Po}(3.2).

Therefore

\begin{align*} \mathrm{P}(C>3)\approx&\,1-e^{-3.2}(1+3.2)\\[4mm] &\,-e^{-3.2}\bigg(\frac{3.2^2}{2!}+\frac{3.2^3}{3!}\bigg)\\[4mm] =&\,0.397480\ldots\\[4mm] =&\,\boxed{0.397}\quad\text{(3 s.f.).} \end{align*}

(b)

解法一

思路

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用本题参数说明大n及小np，满足泊松近似条件。

答题过程

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The sample size is large, $n=200>50$ , and $np=200(0.016)=3.2<5$ , so a Poisson approximation is appropriate.

CIE 9709 2022 November 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