题目

Problem

Nikki is investigating the views of students at her school about the school sports facilities. She plans to give a survey to a sample of students.

Nikki’s friend says, “This survey is about sports facilities, so you should choose a sample of students from the school sports teams.”

(a) State, with a reason, whether you agree with Nikki’s friend.

(1)

Nikki chooses an appropriate random sample of $60$ students. She finds that $45$ of these students think that the sports facilities are good.

(b) Calculate an approximate $95\%$ confidence interval for the proportion of students who think that the sports facilities are good.

(3)

For a different investigation, Nikki uses another large random sample to calculate a $99\%$ confidence interval and an $x\%$ confidence interval.

The width of the $99\%$ confidence interval is double the width of the $x\%$ confidence interval.

(c) Calculate the value of $x$ .

(4)

题目中文翻译

Nikki 正在调查本校学生对学校体育设施的看法。她计划对抽取的一组学生进行问卷调查。

Nikki 的朋友说：“这项调查是关于体育设施的，所以你应该从学校的各支运动队中抽取学生作为样本。”

(a) 说明你是否同意 Nikki 的朋友的看法，并给出理由。

(1)

Nikki 采用适当的随机抽样方法抽取了 $60$ 名学生。她发现，其中有 $45$ 名学生认为体育设施良好。

(b) 计算认为体育设施良好的学生所占比例的近似 $95\%$ 置信区间。

(3)

在另一项调查中，Nikki 使用另一个大容量随机样本，计算出一个 $99\%$ 置信区间和一个 $x\%$ 置信区间。

$99\%$ 置信区间的宽度是 $x\%$ 置信区间宽度的两倍。

(c) 计算 $x$ 的值。

(4)

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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调查对象是全校学生；只抽运动队成员可能产生偏差，不能代表全体学生的意见。

答题过程

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I disagree. Students in sports teams may have different views about the sports facilities from other students, so this sample may not represent all students at the school.

(b)

解法一

思路

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以45/60估计总体比例，并用比例的正态近似标准误建立95%区间。

答题过程

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The sample proportion is $\hat p=\frac{45}{60}=0.75$ . An approximate 95% confidence interval for the population proportion is

\begin{align*} &\,\hat p\pm1.96\sqrt{\frac{\hat p(1-\hat p)}{60}}\\[4mm] =&\,0.75\pm1.96\sqrt{\frac{0.75(0.25)}{60}}\\[4mm] =&\,0.75\pm0.109567\ldots. \end{align*}

Thus the interval is

\boxed{(0.640,\ 0.860)}\quad\text{(3 s.f.).}

(c)

解法一

思路

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两个区间来自同一样本，标准误相同，宽度之比等于正态临界值之比。将99%的临界值减半，再反求中央概率。

答题过程

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Let $s$ be the common estimated standard error for the two intervals, and let $z_x$ be the positive critical value for an $x\%$ confidence interval.

The widths are $2z_{99}s$ and $2z_xs$ , where

z_{99}=\Phi^{-1}(0.995)=2.575829\ldots.

Since the 99% interval has twice the width,

2z_{99}s=2(2z_xs).

Thus

\begin{align*} z_x=&\,\frac{z_{99}}2\\[4mm] =&\,1.287914\ldots. \end{align*}

Consequently

\begin{align*} \frac{x}{100}=&\,2\Phi(z_x)-1\\[4mm] =&\,0.802224\ldots, \end{align*}

so $\boxed{x=80.2}$ (3 s.f.).

CIE 9709 2025 March Paper 62 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