题目

Problem

Bettie believes that $30\%$ of students at her school are vegetarian. Amir claims that the proportion is less than $30\%$ . In order to test his claim, Amir takes a random sample of $20$ students at the school. He finds that exactly $2$ students in the sample say that they are vegetarian.

(a) Use a binomial distribution to test Amir’s claim at the $2.5\%$ significance level.

(5)

(b) State the smallest significance level at which Amir’s claim would be rejected. Give your answer correct to $3$ significant figures.

(1)

题目中文翻译

Bettie 认为她所在学校有 $30\%$ 的学生是素食者。Amir 声称这一比例低于 $30\%$ 。为检验他的说法，Amir 从学校随机抽取了 $20$ 名学生。他发现样本中恰有 $2$ 名学生说自己是素食者。

(a) 用二项分布在 $2.5\%$ 的显著性水平下检验 Amir 的说法。

(5)

(b) 写出会拒绝 Amir 的说法的最小显著性水平。答案保留 $3$ 位有效数字。

(1)

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。

题目勘误

官方原题：(b) 要求会拒绝 Amir 的说法的最小显著性水平。

实际上应为：评分标准给出的临界水平是拒绝原假设的门槛。达到这一门槛时，检验结果支持 Amir 的说法。将显著性水平降到门槛以下，只能得出证据不足以支持其说法。

本页保留官方英文与忠实翻译。

解答

(a)

解法一

思路

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Amir 声称总体比例降低，采用左尾二项检验。应计算至多两名素食者的概率，而不是恰好两名的概率。

答题过程

展开

Let $p$ be the population proportion of vegetarian students. Test

H_0:p=0.3,\qquad H_1:p<0.3.

Let $V$ be the number of vegetarian students in the sample. Under $H_0$ , $V\sim\mathrm B(20,0.3)$ . For the observation $V=2$ , the lower-tail probability is

\begin{align*} \mathrm P(V\le2) =&\,(0.7)^{20}+20(0.3)(0.7)^{19}\\[4mm] &\,\hspace{2pt}+\binom{20}{2}(0.3)^2(0.7)^{18}\\[4mm] \approx&\,0.0354831. \end{align*}

Since $0.0354831>0.025$ , do not reject $H_0$ at the $2.5\%$ significance level.

There is insufficient evidence that the population proportion of vegetarians is less than $30\%$ , so Amir’s claim is not supported at this level.

CIE 9709 2026 June Paper 63 Q2

题目

解答

(a)

解法一

思路

答题过程