题目

Problem

A manufacturer of cell phones claims that $25\%$ of students own a Pumpkin phone. Jeyeraj thinks that the proportion of students at his large college who own a Pumpkin phone is less than $25\%$ . He plans to test the manufacturer’s claim. He chooses a random sample of $30$ students at his college. If the number of students who own a Pumpkin phone is less than $5$ , Jeyeraj will reject the manufacturer’s claim.

(a) State suitable hypotheses for the test.

(1)

(b) Given that the true proportion of students at the college who own a Pumpkin phone is $10\%$ , use a binomial distribution to find the probability of a Type II error.

(3)

At Florence’s college, in a random sample of $40$ students, it was found that $5$ own a Pumpkin phone.

(c) Calculate an approximate $95\%$ confidence interval for the proportion of students at Florence’s college who own a Pumpkin phone.

(3)

题目中文翻译

一家手机制造商声称有 $25\%$ 的学生拥有 Pumpkin 手机。Jeyeraj 认为他所在的大型学院中，拥有 Pumpkin 手机的学生比例低于 $25\%$ 。他打算检验制造商的说法，于是从学院随机抽取了 $30$ 名学生。如果拥有 Pumpkin 手机的学生少于 $5$ 人，Jeyeraj 就会拒绝制造商的说法。

(a) 写出该检验适用的假设。

(1)

(b) 已知该学院拥有 Pumpkin 手机的学生真实比例为 $10\%$ ，用二项分布求发生第二类错误的概率。

(3)

在 Florence 的学院，随机抽取了 $40$ 名学生，发现其中有 $5$ 人拥有 Pumpkin 手机。

(c) 计算 Florence 的学院中拥有 Pumpkin 手机的学生比例的一个近似 $95\%$ 置信区间。

(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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假设参数是该学院的总体持有率，备择是假设比例低于制造商所称的四分之一。

答题过程

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Let $p$ be the population proportion of students at Jeyeraj’s college who own a Pumpkin phone. Suitable hypotheses are

\boxed{H_0:p=0.25,\qquad H_1:p<0.25}.

(b)

解法一

思路

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真实比例改为 $0.1$ ，拒绝规则不变。第二类错误对应样本持有人数至少五人。

答题过程

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Let $K$ be the number of Pumpkin-phone owners in the sample of $30$ . When the true proportion is $0.1$ , $K\sim\mathrm B(30,0.1)$ .

A Type II error occurs when $K\ge5$ , since the rule only rejects $H_0$ when $K<5$ . Hence

\begin{align*} &\,\mathrm P(\text{Type II error})\\[4mm] =&\,1-\sum_{r=0}^4\binom{30}{r}(0.1)^r(0.9)^{30-r}\\[4mm] =&\,1-(0.9)^{30}-30(0.1)(0.9)^{29}\\[4mm] &\,-\binom{30}{2}(0.1)^2(0.9)^{28}\\[4mm] &\,\hspace{2pt}-\binom{30}{3}(0.1)^3(0.9)^{27}\\[4mm] &\,\hspace{4pt}-\binom{30}{4}(0.1)^4(0.9)^{26}\\[4mm] \approx&\,\boxed{0.175}. \end{align*}

(c)

解法一

思路

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这是另一所学院，必须使用 Florence 的五名持有人和四十人样本，不沿用前两问的比例。

答题过程

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For Florence’s college, the sample proportion is $\widehat p=5/40=1/8$ . An approximate $95\%$ confidence interval is

\begin{align*} &\,\widehat p\pm1.96\sqrt{\frac{\widehat p(1-\widehat p)}{40}}\\[4mm] =&\,0.125\pm1.96\sqrt{\frac{(1/8)(7/8)}{40}}\\[4mm] \approx&\,0.125\pm0.102491. \end{align*}

Hence the required interval is $\boxed{(0.0225,\ 0.227)}$ to $3$ significant figures.

CIE 9709 2025 June Paper 61 Q6

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程