题目
When purchasing a pair of glasses from an optician, customers are offered insurance to cover accidental damage.
Past records from the optician show that 30% of customers buy insurance when they purchase a pair of glasses.
In a random sample of 40 customers who have purchased a pair of glasses, represents the number who buy insurance.
(a) State a suitable model for the distribution of
(b) State a necessary assumption for the model in part (a) to be valid.
The probability that fewer than customers who have purchased a pair of glasses buy insurance is less than 0.05
(c) Find the largest possible value of
A second random sample of 200 customers who have purchased a pair of glasses is taken.
Using a normal approximation, the probability that at least of these 200 customers buy insurance is 0.9474, correct to 4 decimal places.
(d) Find the value of . You must show your working.
The optician decided to introduce a special offer in which customers who purchase a pair of glasses get the first three months of insurance free if they buy insurance.
Following this, a random sample of 25 customers who purchase a pair of glasses is taken and 11 of them buy insurance.
(e) Test, at the 10% level of significance, whether or not there is evidence that the proportion of customers who buy insurance when purchasing a pair of glasses has increased. State your hypotheses clearly.
(Total for Question 7 is 14 marks)
题目中文翻译
从眼科医生处购买眼镜时，顾客可获得意外损坏保险。
眼科医生的过往记录显示，30% 的顾客在购买眼镜时会购买保险。
在随机抽取的 40 名已购买眼镜的顾客中， 表示购买保险的人数。
(a) 说明 分布的合适模型。
(b) 说明 (a) 中模型有效所需的一个必要假设。
购买眼镜的顾客中少于 人购买保险的概率小于 0.05
(c) 求 的最大可能值。
又随机抽取了 200 名已购买眼镜的顾客样本。
使用正态近似，这 200 名顾客中至少 人购买保险的概率为 0.9474（精确到 4 位小数）。
(d) 求 的值。你必须展示你的计算过程。
眼科医生决定推出一项特别优惠：购买眼镜的顾客如果购买保险，前三个月免费。
此后，随机抽取了 25 名购买眼镜的顾客样本，其中 11 人购买了保险。
(e) 在 10% 的显著性水平下，检验是否有证据表明购买眼镜时购买保险的顾客比例已经增加。清楚地陈述你的假设。
（第 7 题共 14 分）