题目

Problem

Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number $X$ of days on which she sees it. She models the distribution of $X$ by $\mathrm{B}(31,p)$ , where $p$ is the probability of seeing this kind of bird on a randomly chosen day in July.

Data from previous years suggests that $p=0.3$ , but in $2022$ Rita suspected that the value of $p$ had been reduced. She decided to carry out a hypothesis test.

In July $2022$ , she saw this kind of bird on $4$ days.

(a) Use the binomial distribution to test at the $5\%$ significance level whether Rita’s suspicion is justified.

(5)

In July $2023$ , she noted the value of $X$ and carried out another test at the $5\%$ significance level using the same hypotheses.

(b) Calculate the probability of a Type I error.

(2)

Rita models the number of sightings, $Y$ , per year of a different, very rare, kind of bird by the distribution $\mathrm{B}(365,0.01)$ .

(c)

(i) Use a suitable approximating distribution to find $\mathrm{P}(Y=4)$ .

(3)

(ii) Justify your approximating distribution in this context.

(1)

题目中文翻译

Rita 每年七月都会收集某种鸟的目击数据，作为一个研究项目的一部分。七月里，她每天记下是否看到了这种鸟，并记录看到这种鸟的天数 $X$ 。她用 $\mathrm{B}(31,p)$ 分布描述 $X$ ，其中 $p$ 是在七月随机选取的一天看到这种鸟的概率。

往年的数据表明 $p=0.3$ ，但在 $2022$ 年，Rita 怀疑 $p$ 的值有所下降。她决定进行假设检验。

在 $2022$ 年七月，她有 $4$ 天看到了这种鸟。

(a) 用二项分布在 $5\%$ 的显著性水平下检验 Rita 的怀疑是否有依据。

(5)

在 $2023$ 年七月，她记录了 $X$ 的值，并采用相同的假设，在 $5\%$ 的显著性水平下又做了一次检验。

(b) 计算发生第一类错误的概率。

(2)

Rita 用 $\mathrm{B}(365,0.01)$ 分布描述另一种十分罕见的鸟每年被看到的次数 $Y$ 。

(c)

(i) 用适当的近似分布求 $\mathrm{P}(Y=4)$ 。

(3)

(ii) 结合题意说明所用近似分布为何合适。

(1)

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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检验日观测概率是否下降，使用原假设下二项分布不超过4天的左尾概率。

答题过程

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H_0:p=0.3,\qquad H_1:p<0.3.

Under $H_0$ , $X\sim\mathrm{B}(31,0.3)$ . For the observed value $X=4$ ,

\begin{align*} &\,\mathrm{P}(X\leq4\mid H_0)\\[4mm] =&\,\sum_{r=0}^4\binom{31}{r}(0.3)^r(0.7)^{31-r}\\[4mm] =&\,(0.7)^{31}+31(0.3)(0.7)^{30}\\[4mm] &\,+\binom{31}{2}(0.3)^2(0.7)^{29}\\[4mm] &\,\hspace{2pt}+\binom{31}{3}(0.3)^3(0.7)^{28}\\[4mm] &\,\hspace{4pt}+\binom{31}{4}(0.3)^4(0.7)^{27}\\[4mm] =&\,0.0239034\ldots. \end{align*}

Since $0.0239034\ldots<0.05$ , we reject $H_0$ . There is sufficient evidence at the 5% significance level to support Rita’s suspicion that the probability of seeing this kind of bird on a day in July has decreased.

(b)

解法一

思路

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不能把名义5%直接当作离散检验的实际错误概率。比较相邻左尾概率确定拒绝域，再求原假设下落入拒绝域的概率。

答题过程

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Under $H_0$ , $X\sim\mathrm{B}(31,0.3)$ . From part (a),

\mathrm{P}(X\leq4)=0.0239034\ldots<0.05.

However,

\begin{align*} \mathrm{P}(X\leq5)=&\,0.0239034\ldots\\[4mm] &\,+\binom{31}{5}(0.3)^5(0.7)^{26}\\[4mm] =&\,0.0626628\ldots>0.05. \end{align*}

Thus the rejection region is $X\leq4$ . The probability of a Type I error is

\boxed{0.0239}\quad\text{(3 s.f.).}

(c)(i)

解法一

思路

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罕见事件二项分布用参数np等于3.65的泊松分布近似。

答题过程

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Using $Y\approx\mathrm{Po}(365(0.01))=\mathrm{Po}(3.65)$ ,

\begin{align*} \mathrm{P}(Y=4)\approx&\,e^{-3.65}\frac{3.65^4}{4!}\\[4mm] =&\,0.192214\ldots\\[4mm] \approx&\,\boxed{0.192}\quad\text{(3 s.f.).} \end{align*}

(c)(ii)

解法一

思路

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用题中的具体参数说明试验次数大、每日概率小。

答题过程

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There are many days, $n=365>50$ , and the probability of a sighting on any day is small, $p=0.01<0.1$ , so a Poisson approximation is suitable.

CIE 9709 2024 June Paper 61 Q7

题目

解答

(a)

解法一

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(b)

解法一

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(c)(i)

解法一

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(c)(ii)

解法一

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