科目：統計學與實習
教授：林惠玲
試別：期中考
時間：93年11月16日 (我生日Orz)
by neverneverfu
試題 :
總分：１１０分
一、是非題（18%）（先說明對或錯，錯的請更正，對的請說明理由）
１．要研究台灣的失業率，普查比抽樣調查更容易實施，且較正確。
２．某乙統計學及格的機率是0.3，經濟學及格的機率是0.6，且兩科是否及格彼此無相
關，則其兩科均及格的機率是0.9。
３．某所學校給教師的年薪在30,000到60,000元之間。教師會和學校的董事會正在協議
次年的加薪幅度，假設最後給每位教師加薪５％，則
(1)平均薪資與薪資中位數均增加５％
(2)標準差與變異係數均增加５％
４．陳先生去年拿了100萬委託投顧公司代為買賣股票，去年年底時虧損５０％，今年年
底時賺１００％，因此兩年的平均投資報酬率為２５％。
５．下表是台灣地區大專以上就業者之職業與性別之人數分配表：
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經理人員 非經理人員 單位：千人
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男 189 1520 1709
女 42 1316 1358
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由上表可知擔任經理職務與否與性別有關，且女性受到性別差異歧視。
６．設Ａ、Ｂ、Ｃ三事件獨立，則（Ａ︿Ｂ）與Ｃ為獨立。
_ _
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%）根據行政院主計處調查，台灣地區１５至６４歲已婚女性每天料理家務時間平
均為５小時３０分，標準差為２小時３０分。（資料來源：標準差為估算數字，《台
灣地區婦女婚育與就業調查報告》，行政院主計處，２００１年。）假設已婚女性每
天料理家務時間為常態分配，問：
(1)每天料理家務時間超過８小時者在１５至６４歲已婚女性中所佔比例為何
(2)每個月以３０天計算，則每月料理家務應付酬勞Ｓ（元）與料理家務的時間Ｔ（小
時）有如下的函數關係：Ｓ＝10,000+30．120Ｔ 試問每位已婚女性每月料理家務
應得而未得的平均酬勞與變異數為何？
(3)請求婦女應得家務報酬中位數及Ｑ1、Ｑ3各為何？
三、（10%）台北市每年申報所得稅約有２０萬戶，平均每１０００戶就有１戶計算錯誤，
現抽取１０００戶抽查，另Ｘ為１０００戶中錯誤的份數
(1)則Ｘ為何種機率分配，其機率函數為何？理由。
(2)以近似機率分配計算１０份申報計算有錯誤的機率，並請說明採用近似機率分配的
理由。
四、（12%）某推銷員每天打電話推銷，由過去經驗知平均每５人有１人會購買，若該推銷
員設定的目標為每天推銷３人，因時間有限，每天至多只能打１０通電話，若在１０
通內有３人購買，則隨即停止推銷
(1)另Ｘ為停止推銷的電話通數，則Ｘ的機率函數為何？
(2)是求某天在第８通即停止推銷的機率。
(3)求某天無法達成目標的機率。
五、（12%）請說明可利用哪些方法判斷資料是否為一常態分配？
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(群體) 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?
※未學過微積分者，則可跳答第十題。
學過微積分者，請答第九題，第十題不必答。
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之機率分配函數為：
f(X) = q^x-1 ．p X=1,2,3..... p=0.5 p+q=1
(1)試繪出直方圖。
(2)試求P(X>=6|X>=4)。
(3)試求X之平均數、中位數、眾數、標準差。
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∮完卷∮
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陷自己於忙碌之中 也許不寂寞但卻很孤獨......
陪伴的人也許很多 不孤獨卻不代表不寂寞......
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