题目

Problem

Past evidence shows that 7% of pears grown by a farmer are unfit for sale. This season it is believed that the proportion of pears that are unfit for sale has decreased. To test this belief a random sample of $n$ pears is taken. The random variable $Y$ represents the number of pears in the sample that are unfit for sale.

(a) Find the smallest value of $n$ such that $Y = 0$ lies in the critical region for this test at a 5% level of significance.

(3)

In the past, 8% of the pears grown by the farmer weigh more than 180 g. This season the farmer believes the proportion of pears weighing more than 180 g has changed. She takes a random sample of 75 pears and finds that 11 of them weigh more than 180 g.

(b) Test, using a suitable approximation, whether there is evidence of a change in the proportion of pears weighing more than 180 g. You should use a 5% level of significance and state your hypotheses clearly.

(6)

(Total for Question 4 is 9 marks)

题目中文翻译

过去证据表明，某农场种植的梨中有 7% 不适合出售。本季认为不适合出售的梨的比例下降了。为检验这一判断，随机抽取 $n$ 个梨。随机变量 $Y$ 表示样本中不适合出售的梨的数量。

(a) 求最小的 $n$ 值，使得在 5% 显著性水平下， $Y = 0$ 落入该检验的临界区域。

过去，农场种植的梨中有 8% 重量超过 180 g。本季农场主认为重量超过 180 g 的梨的比例发生了变化。她随机抽取 75 个梨，其中 11 个重量超过 180 g。

(b) 使用适当的近似方法，检验是否有证据表明重量超过 180 g 的梨的比例发生了变化。你应使用 5% 显著性水平并清楚陈述你的假设。

（第 4 题共 9 分）

IAL 2022 June S2 Q4

题目

解答