課程名稱︰統計學上
課程性質︰大二必修
課程教師︰林惠玲
開課系所︰經濟系
考試時間︰2006.10.31. 14:00~17:00
是否需發放獎勵金：是
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統計學 上 第一次小考 林惠玲老師 2006/10/31 PM 2:00～5:00
一、(18%)是非題:(未註明理由不給分)
(1) 某乙統計學及格的機率是0.3,經濟學及格之機率是0.6,且兩科是否及格彼此無關,
則其兩科均及格的機率是0.9.
(2) 陳先生去年拿了100萬元委託投顧公司代為買賣股票,去年年底時虧損50%,今年賺
100%,因此二年的平均投資報酬率為25%
(3) 若兩事件發生的機率皆大於0,且兩事件互斥,則兩事件必不獨立
(4) 若以最小化平方差的觀點,中位數為一組資料最佳的代表值
(5) 只要事件A與事件B不獨立,我們總是能利用事件A發不發生來更新對事件B發生可能
性的預測
(6) 設X為一隨機變數,則 E(1/X) = 1/E(X)
二、(18%)下列資料是一家雜貨店在過去兩週中每日的顧客數:
24,59,80,33,56,75,51,71,81,37,79,78,80,96,40
66,91,86,56,75,53,10,65,74,79,24,81,97,82,66
利用Excel得結果:
------------------------------
欄 1 (a) 請評估該雜貨店顧客數的中央趨勢、分散
------------------------------ 程度、偏度及峰度,並說明資料特性(6%)
平均數 64.83333
標準誤 4.095156 (b) 畫出stem and leaf圖,並說明該每日顧客
中間值 72.5 數分配的情形,又stem and leaf與次數直
眾數 24 方圖有何不同,若畫次數直方圖應分為幾組
標準差 22.4301 較適當,為什麼?(6%)
變異數 503.1092
峰度 -0.01393 (c) 請畫出Box and Whisker plot,並說明如何
偏態 -0.83004 判斷此分配的形狀與特性,或是否有極端值
範圍 87 (outliers) (6%)
最小值 10
最大值 97
總和 1945
個數 30
------------------------------
三、(10%)試證明
(1) E[X-E(X)]^2 ＝ E(X^2)－[E(X)]^2
(2) Σfi‧(Xi-M)^2 ＝Σfi‧Xi^2 － n‧M^2 M為平均數,Xi為i組組中點,n為總個數
四、某大學有二學院,一是工學院,一是文學院,申請入學並不容易,婦女委員會懷疑審核
過程歧視女性。委員會從學校得到以下所有申請者的性別和審核結果的列聯表:
男性（Ｂ） 女性（Ｂ*）
－－－－－－－－－－－－－－－－－－－－－－－
通過（Ａ ） 45 25
不通過（Ａ*） 35 35
－－－－－－－－－－－－－－－－－－－－－－－
總和 80 60
－－－－－－－－－－－－－－－－－－－－－－－
試求
(1) P(A|B),P(A|B*) (6%)
(2) 申請者性別與審核通過有關係嗎? (或獨立嗎?) (5%)
(3) 委員會根據(1)(2)結果,認為學校有性別歧視,若你是該大學校長,你可舉出何
種資料來說明上述資料並不一定代表性別歧視 (4%)
五、(15%)設A,B二支球隊進行五戰三勝的比賽,每場球A勝之機率為0.6,且前勝負無關.
(a) A贏得比賽的機率? B贏得比賽的機率?
(b) 比賽打到第五場才分出勝負的機率?
(c) A一路領先,從未被追平的機率?
六、(16%)The figure shown below is a representation of a comprised of two
subsystems that are said to operate in parallel. Each subsystem has two
components that operate in series. The system will operate properly as
long as at least one of the subsystems functions properly. The probability
of failure of each component in the system (eg. component #1) is 0.1.
Assume the components operate independently of each other.
A system comprised of two parallel subsystems
Subsystem A
#1 ------------→ #2
Input → ↗ Subsystem B ↘ → Output
↘ #3 ------------→ #4 ↗
(a) Find the probablilty that the system operates properly.
(b) Find the probability that exactly one subsystems fails.
(c) Find the probability that the system fails to operate properly.
(d) How many parallel subsystems like the two shown here would be required
to guarantee that the system would operate properly at least 99% of
the time?
七、(8%) A diagnostic test for a disease is said to be 99% 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,
(a) What is the conditional probability that she does,in fact, have
the disease? (5%)
(b) Are you surprised by the answer? Would you call this diagnostic
test reliable? Explain it. (3%)
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