题目

Problem

A student wishes to estimate the proportion, $p$ , of students at her college who have exactly one brother. She surveys a random sample of $50$ students at her college and finds that $18$ of them have exactly one brother. She calculates an approximate $a\%$ confidence interval for $p$ and finds that the lower limit of the confidence interval is $0.244$ correct to $3$ significant figures.

Find $a$ correct to the nearest integer.

(4)

题目中文翻译

一名学生想估计她所在学院恰有一名兄弟的学生所占的比例 $p$ 。她从学院随机抽取 $50$ 名学生进行调查，发现其中 $18$ 人恰有一名兄弟。她计算了 $p$ 的近似 $a\%$ 置信区间，发现区间的下限为 $0.244$ ，精确到 $3$ 位有效数字。

求 $a$ ，答案精确到最接近的整数。

(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

解法一

思路

展开

以样本比例0.36写出下限公式，反求正态临界值，再由中央概率求信心水平并按要求取整数。

答题过程

展开

The sample proportion is $\hat p=\frac{18}{50}=0.36$ . Using the stated lower limit,

0.36-z\sqrt{\frac{0.36(0.64)}{50}}=0.244.

Hence

z=\frac{0.36-0.244}{\sqrt{0.36(0.64)/50}} =1.708841\ldots.

The central confidence probability is

\frac{a}{100}=2\Phi(z)-1=0.9125197\ldots.

Thus $a=91.25197\ldots$ , giving $\boxed{a=91}$ to the nearest integer.

CIE 9709 2024 June Paper 62 Q3

题目

解答

解法一

思路

答题过程