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CIE 9709 2023 June Paper 62 Q1

A Level / CIE / S2

CIE 9709 2023 June Paper 62 Paper · Question 1

题目

Problem

In a survey of 200200 randomly chosen students from a certain college, 23%23\% of the students said that they owned a car.

Calculate an approximate 93%93\% confidence interval for the proportion of students from the college who own a car.

(3)
题目中文翻译

从某学院随机抽取 200200 名学生进行调查，其中 23%23\% 的学生表示自己拥有汽车。

计算该学院拥有汽车的学生所占比例的近似 93%93\% 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

解法一

思路

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样本比例为0.23，93%中央置信概率对应两端各3.5%，求正态临界值后建立比例区间。

答题过程

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With p^=0.23\hat p=0.23 and z=Φ−1(0.965)=1.811911…z=\Phi^{-1}(0.965)=1.811911\ldots, the approximate 93% confidence interval is

0.23±z0.23(0.77)200=0.23±0.0539177….0.23\pm z\sqrt{\frac{0.23(0.77)}{200}} =0.23\pm0.0539177\ldots.

Thus the interval for the population proportion is

(0.176, 0.284)(3 s.f.).\boxed{(0.176,\ 0.284)}\quad\text{(3 s.f.).}