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IAL 2026 Jan S2 A Q4

A Level / Edexcel / S2

IAL 2026 Jan A Paper · Question 4

题目

Problem

A company claims that 35% of its peas germinate. In order to test this claim Ann decides to plant 15 of these peas and record the number which germinate.

(a) (i) State suitable hypotheses for a two tailed test of this claim.

(ii) Using a 5% level of significance, find an appropriate critical region for this test. The probability in each of the tails should be as close to 2.5% as possible.

(4)

(b) Ann found that 8 of the 15 peas germinated. State whether or not the company’s claim is supported. Give a reason for your answer.

(2)

(c) State the actual significance level of this test.

(1)

(Total for Question 4 is 7 marks)

题目中文翻译

一家公司声称其 35% 的豌豆会发芽。为了检验这一说法，Ann 决定种植 15 颗这样的豌豆并记录发芽的数量。

(a) (i) 为此双尾检验陈述合适的假设。

(ii) 使用 5% 的显著性水平，为此检验找到合适的临界区域。每尾的概率应尽可能接近 2.5%。

(b) Ann 发现 15 颗豌豆中有 8 颗发芽。说明公司的说法是否得到支持。给出你的理由。

(c) 说明此检验的实际显著性水平。

（第 4 题共 7 分）

解答

(a)(i)

解法一

思路

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令 pp 表示一颗该公司豌豆发芽的概率。公司的声称作为原假设；题目要求双尾检验，因此备择假设表示发芽概率不等于 0.350.35。

答题过程

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Let pp be the probability that a pea germinates. The hypotheses are

H0:p=0.35\boxed{H_0:p=0.35}

and

H1:p≠0.35.\boxed{H_1:p\neq0.35}.

(a)(ii)

解法一

思路

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在 H0H_0 下，发芽颗数 XX 服从参数为 1515 和 0.350.35 的二项分布。双尾检验要求每一尾的概率尽量接近但不超过 0.0250.025；分别比较下尾与上尾相邻整数边界，确定临界区域。

答题过程

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Under H0H_0,

X∼B⁡(15,0.35).X\sim\operatorname{B}(15,0.35).

For the lower tail,

P(X≤1)=0.01418,P(X≤2)=0.06173.P(X\leq1)=0.01418, \qquad P(X\leq2)=0.06173.

For the upper tail,

P(X≥10)=0.01244,P(X≥9)=0.04219.P(X\geq10)=0.01244, \qquad P(X\geq9)=0.04219.

Thus the probability in each tail is as close to 0.0250.025 as possible without exceeding it. The critical region is

X≤1orX≥10.\boxed{X\leq1\quad\text{or}\quad X\geq10}.

(b)

解法一

思路

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把观察值 X=8X=8 与 (a)(ii) 的临界区域比较。它不在临界区域内，因此没有足够证据拒绝公司的声称，结果支持该声称。

答题过程

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The observed value X=8X=8 does not lie in the critical region. Therefore, H0H_0 is not rejected. There is insufficient evidence that the germination rate differs from 35%35\%, so the company’s claim is supported.

(c)

解法一

思路

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离散分布的实际显著性水平等于原假设成立时落入整个临界区域的概率，即两个尾部概率之和。

答题过程

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The actual significance level is

P(X≤1)+P(X≥10)= 0.0141788+0.0124432= 0.026622.\begin{align*} P(X\leq1)+P(X\geq10) =&\,0.0141788+0.0124432\\ =&\,0.026622. \end{align*}

Therefore, the actual significance level is

2.66%.\boxed{2.66\%}.