题目

Problem

A biased spinner has four sides. Each side is of a different colour: yellow, red, green or black. The probability, $p$ , that the spinner will land on red is unknown. The spinner was spun $200$ times, and the proportion, $a$ , of times that it landed on red was noted. This proportion was used to calculate an approximate $90\%$ confidence interval for $p$ . The width of this confidence interval was $0.1066$ correct to $4$ significant figures.

Find the two possible values of $a$ .

(4)

题目中文翻译

一个不均匀的陀螺有四个面，颜色各不相同，分别为黄色、红色、绿色和黑色。陀螺落在红色面的概率 $p$ 未知。将陀螺转动 $200$ 次，记录落在红色面的次数所占的比例 $a$ 。用该比例计算出 $p$ 的一个近似 $90\%$ 置信区间。该区间的宽度保留 $4$ 位有效数字后为 $0.1066$ 。

求 $a$ 的两个可能值。

(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

解法一

思路

展开

$a$ 是样本比例而不是人数。用区间宽度反求 $a(1-a)$ ，再解两根并核对四舍五入精度。

答题过程

展开

The approximate $90\%$ interval has width

2(1.645)\sqrt{\frac{a(1-a)}{200}}.

Using the given width,

\begin{align*} a(1-a) \approx&\,200\bigg(\frac{0.1066}{3.29}\bigg)^2\\[4mm] \approx&\,0.209968,\\[4mm] a^2-a+0.209968\approx&\,0. \end{align*}

Solving gives the two possible proportions, to $3$ significant figures,

\boxed{a=0.300\quad\text{or}\quad a=0.700}.

These correspond to $60$ or $140$ red outcomes out of $200$ spins. Either gives a width approximately $0.106608$ , which rounds to $0.1066$ to $4$ significant figures.

CIE 9709 2025 June Paper 62 Q4

题目

解答

解法一

思路

答题过程