課程名稱︰統計學
課程性質︰經濟二必修
課程教師︰林惠玲
開課學院：社科院
開課系所︰經濟系
考試時間︰2006.11.21
是否需發放獎勵金：是
試題 :
統計學上 期中考試 林惠玲老師
95.11.21 考試時間: 14:20~17:00 滿分107分
一、選擇題(答案可能不只一個,每題2分,共10分)
1. Which of the following statements are true?
(a) The variance will always be larger than the mean absolute the deviation.
(b) The variance can never be negative.
(c) The larger the variance is for a set of data,the more meaningful the mean
is.
(d) The larger the variance is for a set of data, the less meaningful the mean
is.
2. Which of the following statement is true?
(a) When grouping data, the number of classes is unimportant.
(b) When the Lorenz curve is a straight line, which is perfect equality of
income.
(c) Information may be lost when raw data are grouped.
(d) A stem-and-leaf display can be used for servicing very large count data.
3. The difference between the binomial distribution and the hypergeometric
distribution is that
(a) The binomial distribution assumes that the probability of success is
constant, whereas the hypergeometric distribution assumes that it changes
with each trial.
(b) The binomial distribution assumes that the probability of success changes,
whereas the hypergeometric distribution assumes that it remains constant.
(c) The variance of hypergeometric distribution is larger than that of binomial
distribution.
(d) The variance of hypergeometric distribution is smaller than that of binomial
distribution.
4. 若A、B定義為在樣本空間(sample space)的兩個事件，則:
_ _ _
(a) P(A∪B)=P(A)-P(A∩B) (b) P(A|B)+P(A|B)=1
_ _ _
(c) 若A、B獨立，則A、B獨立 (d) P(A|B)+P(A|B)=1
5. 下列的敘述何者為正確?
(a) 若X與Y分別為成功機率為P1,P2兩個獨立的二項分配，
定義隨機變數W=X+Y，W亦必為二項分配
(b) 若X與Y分別為平均數λx,λy 的兩個獨立的Poisson分配，
定義W=X+Y，Y亦為成功機率P的二項分配
(C) 若X為成功機率P的二項分配，義Y=n-X，Y亦為成功機率P的二項分配
(d) 設X為Bernoulli分配，則E(X)恰為成功機率P，V(X)為pq/2
二、(28分) 是非題(請先說明「是」或「非」，再明理由或證明，未說明理由不予計分)
1. 設X為一隨機變數，其平均為μ,變異數為σ^2，則：
2 2
(a) Ｅ ( 2Ｘ ＋ 3Ｘ ＋ 5 ）= 2μ + 3μ + 5
2 4 2
(b) Ｖ ( 2Ｘ ＋ 3Ｘ ＋ 5 ）= 4σ + 9σ
2. 某電子公司舉行尾牙，並進行摸彩，但並非通通有獎，員工共200人，獎品為5張股票，
共有20人可中獎。某員工認為先抽與後抽中獎的機率是一樣的。
_
3. 設A、B為互斥事件(mutually exclusive events)，P(A|AUB)=P(A)/[P(A)+P(B)]
2 2
4. 設X1為常態分配，其平均數為μ，變異數為σ ，即X1～N(μ，σ )
X2=2X1，Y=X1+X2
2 2
(a)X2～N(2μ，2σ ) (b)Y～N(3μ，9σ )
5. 以下20位同學統計學期末考成積，此組資料沒有Outliers
97 63 20 88 74 70 58 68 73 81
84 92 77 62 65 76 76 72 65 61
三、(9分) 在檢驗一批產品的過程中，觀察出直到出現兩個不良品為止，現若令隨機變數
x表示此檢驗過程出現的良品個數，並設不良品的機率為ｐ，試問
(a) 該隨機實驗的樣本空間,並列出x的值
(b) 求出x的機率分配
(c) p=0.1,計算p(x<2)
四、(8分)某電腦製造商所生產的電腦的使用年數為一常態分配，根據調查，己知產品中
有２０％的電腦使用時間低於三年，有１０％的電腦超過七年，求此廠商所生產的
電腦之使用年數分配的平均數與標準差
五、(6分)To reduce theft, the Meredeth Company screens all its empolyees
with a lie detector test that is known to be corret 90 percent of the
time (for both guilty and innocent subjects). George Meredeth decides
to fire all employees who fail the test. Suppose 5 percent of the employees
are guilty of theft.
(a) Of the workers fired, what proportion is actually guilty?
(b) What do you think of George's policy?
六、(10分)Suppose that on averate 35 phone calls reached at a certain
residence per week:
(a) Find the probability that the phone rings k times, k=1,2,..., on a
certain day? State the reasons for you model.
(b) If all the members of the residence plan to go out shopping, how long
can they be away if they wish the probability of not missnig any phone
calls to be at least 0.5?
七、(12分)Consider a multiple-choice exaination with 50 questions. Each
question has 4 possible answers. Assume that a student who has done the
homework and attended lectures has a 75 possibility of answering any
question correctly.
(a) A student must answer 30 or more questions correctly to pass the
examination. What percentage of the students who have done their
homework and attended lectures will pass the examination?
(b) Assume that a student has not attended class and has not the homework
for the course. Furthermore, assume that the student will simple
guess at the answer to each question. What is the possibility that
this student will answer 30 or more questions correctly and pass the
examination?
(c) A student must answer 43 or more questions correctly to obtain grade
of A. What percentage of the students who have done their homework and
attended lectures will obtain grade of A on this multiple-choice
examination? If a student wants to increase the probability to obtain
grade A, what should he do?
經濟系及修過微積分的外系同學請答第八題.其他同學請跳答第九題.
八、(18分)The length of time required by students to complete an 1-hour exam
is a random variable with a density function given by
f(y)=cy^2+y, 0≦y≦1
f(y)=0, elsewhere
(a) Find c.
(b) Graphy f(y)
(c) Find the probability that a randomly selected student will finish in
less than half an hour.
(d) Given that a particular student needs at least 15 minutes to complete
the exam, find the probability that she will require at least 30
minutes to finish?
(e) Find the mean and variance of Y
九、(18分) 已知某工廠100名員工每月薪資(x)、教育程度之機率分配表如下:
┌──────┬──────┬─────┬─────┬─────┐
│ 薪資 (萬元)│ 1～2 │ 2～3 │ 3～4 │ 4～5 │
├──────┼──────┼─────┼─────┼─────┤
│ 機率 │ 0.20 │ 0.35 │ 0.30 │ 0.15 │
├──────┼──────┼─────┼─────┼─────┤
│大專以上(人)│ 5 │ 12 │ 17 │ 12 │
├──────┼──────┼─────┼─────┼─────┤
│未達大專(人)│ 15 │ 23 │ 13 │ 3 │
└──────┴──────┴─────┴─────┴─────┘
(a) 求工廠員工平均薪資水準E(X)及變異數V(X),薪資以組中點計算
(b) 若政府課稅是10,000元以下免稅,而10,000元以上部分一律課20%的比例稅,
請求出該工廠員工的稅後平均薪資水準及變異數。
(c) 若從員工中任意抽取一人，則此人為未達大專教育程度條件下，薪資為
4～5萬元的機率為何?
(d) 請說明教育程度與薪資是否有關
十、(7分)請証明柴比氏定理(Chevbyshev's theorem)
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