题目

Problem

The number of cars arriving at a certain road junction on a weekday morning has a Poisson distribution with mean $4.6$ per minute. Traffic lights are installed at the junction and a council officer wishes to test at the $2\%$ significance level whether there are now fewer cars arriving. He notes the number of cars arriving during a randomly chosen $2$ -minute period.

(a) State suitable null and alternative hypotheses for the test.

(1)

(b) Find the critical region for the test.

(4)

The officer notes that, during a randomly chosen $2$ -minute period on a weekday morning, exactly $5$ cars arrive at the junction.

(c) Carry out the test.

(2)

(d) State, with a reason, whether it is possible that a Type I error has been made in carrying out the test in part (c).

(1)

The number of cars arriving at another junction on a weekday morning also has a Poisson distribution with mean $4.6$ per minute.

(e) Use a suitable approximating distribution to find the probability that more than $300$ cars will arrive at this junction in an hour.

(3)

题目中文翻译

工作日上午，到达某个路口的汽车数服从泊松分布，平均每分钟 $4.6$ 辆。该路口安装了交通信号灯，一名地方议会的工作人员想在 $2\%$ 的显著性水平下检验现在到达的汽车数是否减少。他随机选取一段 $2$ 分钟的时间，记录到达的汽车数。

(a) 写出适合该检验的原假设和备择假设。

(1)

(b) 求该检验的拒绝域。

(4)

该工作人员在一个工作日的上午随机选取一段 $2$ 分钟的时间，记录到恰好有 $5$ 辆汽车到达路口。

(c) 进行该检验。

(2)

(d) 说明 (c) 的检验中是否可能犯了第一类错误，并给出理由。

(1)

工作日上午，到达另一个路口的汽车数也服从泊松分布，平均每分钟 $4.6$ 辆。

(e) 用适当的近似分布求一小时内到达这个路口的汽车超过 $300$ 辆的概率。

(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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以安装交通灯后的每分钟总体平均到达数为参数，备择为减少。

答题过程

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Let $\lambda$ be the population mean number of cars arriving per minute on weekday mornings after the traffic lights were installed.

\boxed{H_0:\lambda=4.6,\qquad H_1:\lambda<4.6}.

(b)

解法一

思路

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原假设下两分钟总数参数为9.2。比较相邻左尾概率与2%，找到最大的拒绝边界。

答题过程

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Let $C$ be the number arriving in two minutes. Under $H_0$ , $C\sim\mathrm{Po}(9.2)$ .

\begin{align*} \mathrm{P}(C\leq3)=&\,e^{-9.2}\bigg(1+9.2+\frac{9.2^2}{2!}\bigg)\\[4mm] &\,+e^{-9.2}\frac{9.2^3}{3!}\\[4mm] =&\,0.0184196\ldots<0.02. \end{align*}

However,

\begin{align*} \mathrm{P}(C\leq4)=&\,\mathrm{P}(C\leq3)+e^{-9.2}\frac{9.2^4}{4!}\\[4mm] =&\,0.0485796\ldots>0.02. \end{align*}

Therefore the critical region is $\boxed{C\leq3}$ .

(c)

解法一

思路

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5辆不在拒绝域，因而不拒绝原假设并给出不足证据的结论。

答题过程

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The observed count of 5 is outside the critical region $C\leq3$ , so $H_0$ is not rejected. There is insufficient evidence at the 2% significance level that the population mean arrival rate has decreased.

(d)

解法一

思路

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第一类错误必须拒绝真实原假设；本次未拒绝。

答题过程

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No. A Type I error requires rejection of a true $H_0$ , whereas $H_0$ was not rejected in part (c).

(e)

解法一

思路

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一小时总数参数为276，正态近似并将超过300修正为300.5。

答题过程

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For the one-hour total,

\begin{align*} C_{60}\sim&\,\mathrm{Po}(60(4.6))=\mathrm{Po}(276)\\[4mm] \approx&\,\mathrm{N}(276,276). \end{align*}

With a continuity correction,

\begin{align*} \mathrm{P}(C_{60}>300)\approx&\,1-\Phi\bigg(\frac{300.5-276}{\sqrt{276}}\bigg)\\[4mm] =&\,\boxed{0.0701}\quad\text{(3 s.f.).} \end{align*}

CIE 9709 2022 June Paper 62 Q4

题目

解答

(a)

解法一

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(b)

解法一

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(c)

解法一

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(d)

解法一

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(e)

解法一

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