科目：统计学与实习 教授：林惠玲 试别：期中考 时间：93年11月16日　(我生日Orz) by neverneverfu 试题 : 总分：１１０分 一、是非题（18%）（先说明对或错，错的请更正，对的请说明理由） 　１．要研究台湾的失业率，普查比抽样调查更容易实施，且较正确。 　２．某乙统计学及格的机率是0.3，经济学及格的机率是0.6，且两科是否及格彼此无相 　　　关，则其两科均及格的机率是0.9。 　３．某所学校给教师的年薪在30,000到60,000元之间。教师会和学校的董事会正在协议 　　　次年的加薪幅度，假设最後给每位教师加薪５％，则 　　　(1)平均薪资与薪资中位数均增加５％ 　　　(2)标准差与变异系数均增加５％ 　４．陈先生去年拿了100万委托投顾公司代为买卖股票，去年年底时亏损５０％，今年年 　　　底时赚１００％，因此两年的平均投资报酬率为２５％。 　５．下表是台湾地区大专以上就业者之职业与性别之人数分配表： 　　　 ─────────────────── 　　　　　　　　　　　　经理人员　　　非经理人员 单位：千人 ─────────────────── 　　　　　　　　　　男　　189 1520 1709 女 42 1316 1358 ─────────────────── 　　　由上表可知担任经理职务与否与性别有关，且女性受到性别差异歧视。 　６．设Ａ、Ｂ、Ｃ三事件独立，则（Ａ︿Ｂ）与Ｃ为独立。 _ _ 　７．If A and B are two events,then P（Ａ︿Ｂ）>= 1-P(A)-P(B) ８．Let X and Y be Binomial random variables with X~B(n1,P1),Y~B(n2,P2), then X+Y is binomial(n1+n2,P1+P2) 二、（15%）根据行政院主计处调查，台湾地区１５至６４岁已婚女性每天料理家务时间平 　　均为５小时３０分，标准差为２小时３０分。（资料来源：标准差为估算数字，《台 　　湾地区妇女婚育与就业调查报告》，行政院主计处，２００１年。）假设已婚女性每 　　天料理家务时间为常态分配，问： 　　(1)每天料理家务时间超过８小时者在１５至６４岁已婚女性中所占比例为何 (2)每个月以３０天计算，则每月料理家务应付酬劳Ｓ（元）与料理家务的时间Ｔ（小 　　　 时）有如下的函数关系：Ｓ＝10,000+30．120Ｔ　试问每位已婚女性每月料理家务 　　　 应得而未得的平均酬劳与变异数为何？ (3)请求妇女应得家务报酬中位数及Ｑ1、Ｑ3各为何？ 三、（10%）台北市每年申报所得税约有２０万户，平均每１０００户就有１户计算错误， 　　现抽取１０００户抽查，另Ｘ为１０００户中错误的份数 　　(1)则Ｘ为何种机率分配，其机率函数为何？理由。 (2)以近似机率分配计算１０份申报计算有错误的机率，并请说明采用近似机率分配的 　　　 理由。 四、（12%）某推销员每天打电话推销，由过去经验知平均每５人有１人会购买，若该推销 　　员设定的目标为每天推销３人，因时间有限，每天至多只能打１０通电话，若在１０ 　　通内有３人购买，则随即停止推销 　　(1)另Ｘ为停止推销的电话通数，则Ｘ的机率函数为何？ (2)是求某天在第８通即停止推销的机率。 (3)求某天无法达成目标的机率。 五、（12%）请说明可利用哪些方法判断资料是否为一常态分配？ 六、（9%）Product reliability has been defined as the probability that a product will perform its intended function satisfactorily for its intended life when operating under specified conditions. The reliability function, R(x),for a product indicates the probility of the product's life exceeding x time periods. When the time until failure of a product can be adequately modeled by an exponential distribution,theproduct's reliability function is R(x)=ｅ^-λx (Ross,Stochastic Process,1996).Suppose that the time to failure(in years) of a particular product is modeled by an exponential distribution with λ=0.2. (1)What is the probability that the product will perform satisfactorily for at least four years? (2)If λ changes,will the probability that you caculated in part (1) change? Explain. (3)How long should the length of the warranty period be for the product if the manufacturer wants to replace no more than 5% of the units sold while under warranty? 七、（10%）The number of bacteria colonies(群体)　of a certain type in samples of polluted water has a Poisson distribution with a mean of 2 per cubic centimeter. (1)If four 1-cubic-centimeter sample are independently selected from this water,find the probability that at least one sample will contain one or more bacteria colonies. (2)How many 1-cubic-centimeter samples should be selected in order to have a probability of approximately 0.95 of seeing at least one bacteria colony? 八、（8%）A diagnostic test for a disease is said to be 90% accurate in that if a person has the disease,the test will detect it with probability 0.9. Also,if a person does not have the disease,the test will report that he or she does not have it with probability 0.9.Only 1% of the population has the disease in question.If a person is chosen at random from the population and the diagnostic test indicates that she has the disease,what is the conditional probability that she does,in fact,have the disease?Are you surprised by the answer?Would you call this diagnosttic test reliable? How do you improve the reliability of this diagnostic test? ※未学过微积分者，则可跳答第十题。 　学过微积分者，请答第九题，第十题不必答。 九、（16%）A supplier of kerosene has a 150-gallon tank that is filled at the begining of each week.His weekly demand shows a relative frequency behavier that increases steadily up to 100 gallons and then levels off between 100 and 150 gallons.If Y denotes weekly demand in hundreds of gallons,the relative frequency of demand can be modeled by y, 0 < = y < = 1 f(y) = 1, 1 < y < = 1.5 0, elsewhere. (1)Graph f(y). (2)Given that the weekly demand more than 100gallons,find the probability that the weekly demand more than 120 gallons. (3)Find the mean and variance of Y. 十、（16%）若Ｘ之机率分配函数为： 　　f(X) = q^x-1　．p X=1,2,3..... p=0.5 p+q=1 (1)试绘出直方图。 　　(2)试求P(X>=6|X>=4)。 (3)试求X之平均数、中位数、众数、标准差。 ---------------------------------------------------------------------- ∮完卷∮ -- 陷自己於忙碌之中　也许不寂寞但却很孤独...... 陪伴的人也许很多　不孤独却不代表不寂寞...... --

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