世新大學八十八學年度碩士班招生考試

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考試科目

經濟學系碩士班

總體經濟學

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Please answer each of the following problems in sequence using either English or Chinese. The scores are given for each problem. You may use your own pocket calculator for needed computations.

 

1.      Two cards are drawn at random without replacement from an ordinary deck. Let X be the number of hearts and Y the number of black cards obtained.（本題10分）

(a)    Write an expression for the joint pdf, f (x,y).

(b)   Find the marginal pdf’s, f1(x) and f2(y).

(c)    Are X and Y independent?

(d)   Find P [Y=1|X=1].

(e)    Find P [Y=y|X=x].

 

2.      A book publisher ships 100 books to a dealer. Each book costs 25ψto produce, and the publisher receives $1.00 for each book that sells. The unsold books are discarded. If the demand for the books follows a binomial distribution D~BIN (100,p); （本題10分）

(a)    Find the expected profit for the publisher.

(b)   How large must p be to produce s positive expected profit?

(c)    Suppose instead that D~POI (25). Express the expected profit in this case.

(d)   Would the publisher make more expected profit by shipping more books if D~POI (25)?

 

3.      Suppose that Xi~N (μ,σ2), i = 1,…,n, and Zi~N(0,1), i = 1,…, k, and all variables are independent.

State the distribution of each of the following variables and write down their associated parameter(s) (no proof required) （本題10分）

(a) X2+2X3    (b) (Z1)2+(Z2)2    (c) Z1/[(Z2)2]1/2    (d) (Z1)2/(Z2)2    (e) kZ2.

 

4.      Please briefly explain the following statistical terms: (本題10分)

(a) Empirical Rule    (b) Central Limit Theorem    (c) Hypergeometric Distribution

(d) Statistic          (e) Confidence Interval

 

5.      The owner of the Franklin Park Mall are studying the shopping habits of their customers. From earlier studies the owners are under the impression that a typical shopper spends 0.75 hours at the mall with a standard deviation of 0.10 hours. Recently the mall owners have added some specialty restaurants designed to keep shoppers in the mall longer. A consulting firm, Brunner and Swanson Marketing Enterprises, has been hired to evaluate the effects of the restaurants. A sample of 45 shoppers by Brunner and Swanson revealed that the mean time spend in the mall had increased to 0.80 hours. （本題10分）

(a)    Develop a test of hypothesis to determine if the mean time spend in the mall is more than 0.75 hours. Use the 0.05 significance level.

(b)   Suppose the mean shopping time actually increased from 0.75 hours to 0.77 hours. What is the probability this increase would not be detected?

(c)    When Brunner and Swanson reported the information in part (b) to the mall owners, they were upset with the statement that a survey could not detect a change from 0.75 to 0.77 hours of shopping time. How could this probability be reduced?


6. Shank’s, Inc., a nationwide advertising firm, wants to know if the size of an advertisement and the color of the advertisement make a difference in the response of magazine readers. A random sample of readers is shown ads of four different colors and three different sizes. Each reader is asked to give the particular combination of size and color a rating between 1 and 10. Assume that the ratings are approximately normally distributed. The rating for each combination is shown in the following table (for example, the rating for a small red ad is 2). (Assume no interaction effect of size and color). （本題10分）

                                  COLOR OF AD

SIZE OF AD

RED

BLUE

ORANGE

GREEN

SMALL

2

3

3

8

MEDIUM

3

5

6

7

LARGE

6

7

8

8

  

(a)    Is there a difference in the effectiveness of an advertisement by color? (Use the 0.05 level of significance). (F3,6,0.05 = 4.76)

(b)   Is there a difference in the effectiveness of an advertisement by size? (Use the 0.05 level of significance). (F2,6,0.05 = 5.14)

 

7.          考慮一簡單線性迴歸模型，並回答下列問題（共40分）：

           Yt＝α+βXt +μt, t ＝1, 2,..., T ,μt ~ N (0, σ2)

（1）因變數（Yt）、自變數（Xt）和誤差項（μt），何者為隨機變數？何者為非隨機變數？（4分）

（2）誤差項（μt）為什麼會存在，它可能包含哪些因素或效果？若μt ~ N (0, σt2)，則此一迴歸式可能產生何種統計問題？（4分）

（3）試求迴歸係數α和β的最小平方估計式？（4分）

（4）試證明。(4分)

（5）若β改用估計，則該估計式否具不偏性？（4分）

（6）若Yt為消費支出（單位：元），Xt為所得（單位：元），則上式為一典型的消費函數。若某甲以台灣地區1953∼1999年資料估得一簡單迴歸線，並得出判定係數（R2）為0.36，請問你（妳）如何解釋0.36的意義？（4分）

（7）接題（6）的相關係數為多少? (4分)

（8）接題（6），今將消費支出和所得改用千元為單位，重新估計，則迴歸係數α和β會如何改變？（4分）

（9）若將因變數（Yt）和自變數（Xt）標準化，重新禿記，試證明迴歸係數β恰為Yt與Xt的樣本相關係數。（4分）

（10）若將Yt改為自變數，並將Xt改為因變數，重新估計，在何條件下，迴歸係數β會相同。（4分）