题目
Luis makes and sells rugs. He knows that faults occur randomly in his rugs at a rate of 3 every 4 m^2
(a) Find the probability of there being exactly 5 faults in one of his rugs that is 4 m^2 in size.
(b) Find the probability that there are more than 5 faults in one of his rugs that is 6 m^2 in size.
Luis makes a rug that is 4 m^2 in size and finds it has exactly 5 faults in it.
(c) Write down the probability that the next rug that Luis makes, which is 4 m^2 in size, will have exactly 5 faults. Give a reason for your answer.
A small rug has dimensions 80 cm by 150 cm. Faults still occur randomly at a rate of 3 every 4 m^2
Luis makes a profit of £80 on each small rug he sells that contains no faults but a profit of £60 on any small rug he sells that contains faults.
Luis sells n small rugs and expects to make a profit of at least £4000
(d) Calculate the minimum value of n
Luis wishes to increase the productivity of his business and employs Rhiannon. Faults also occur randomly in Rhiannon’s rugs and independently to faults made by Luis.
Luis randomly selects 10 small rugs made by Rhiannon and finds 13 faults.
(e) Test, at the 5% level of significance, whether or not there is evidence to support the suggestion that the rate at which faults occur is higher for Rhiannon than for Luis. State your hypotheses clearly.
(Total for Question 2 is 15 marks)
题目中文翻译
Luis 制作并销售地毯。他知道自己地毯上的瑕疵是随机出现的，速率为每 4 m^2 有 3 个瑕疵。
(a) 求一块 4 m^2 地毯上恰好有 5 个瑕疵的概率。
(b) 求一块 6 m^2 地毯上瑕疵多于 5 个的概率。
Luis 做了一块 4 m^2 的地毯，发现其中恰好有 5 个瑕疵。
(c) 写下 Luis 接下来做的一块 4 m^2 地毯恰好有 5 个瑕疵的概率，并给出理由。
一块小地毯的尺寸为 80 cm × 150 cm。瑕疵仍以每 4 m^2 有 3 个的速率随机出现。
Luis 每卖出一块没有瑕疵的小地毯可赚 £80；每卖出一块有瑕疵的小地毯可赚 £60。
Luis 卖出 n 块小地毯，并期望总利润至少为 £4000。
(d) 求 n 的最小值。
Luis 想提高业务效率，于是雇佣了 Rhiannon。Rhiannon 的地毯也随机出现瑕疵，而且与 Luis 的瑕疵相互独立。
Luis 随机抽查了 Rhiannon 制作的 10 块小地毯，发现共有 13 个瑕疵。
(e) 在 5% 显著性水平下检验是否有证据支持“Rhiannon 的瑕疵发生率高于 Luis”的说法，并清楚写出假设。
（第 2 题共 15 分）