109年 - [非官方正解]109 國立高雄大學_碩士班招生考試_經營管理研究所：統計學#103280

I. (60％)MULTIPLE CHOICE QUESTION
 所有的答案請寫在答案卷，答案寫法如下列所示: All answers must be written on the answer sheet for example:
  
  1. Which of the following statement on a simple OLS regression functionis true ? (A) Coefficient of determinationR ²  is equal to the correlation coefficient of the sample value X and Y . (B) More samples necessarily leads to a lower (C)is always greater than (D)Under the same significant level, if is significantly greater than zero, then is also significantly greater than zero.

6. N samples is randomly selected from a normal distribution population with (A)σ ²  (B)  μ² (C)σ²+μ²(D)

7. Along with 6, if we have two estimatorsX to estimate μ , which of the following statement is true? (A) Both are unbias. (B) Onlyis asymptotically unbias. (C) is more efficient than. (D)Only is a consistent estimator.

8. If random variable X is uniformly distributed between 2 and 4 , then Var (X) = (A) (B)(C)  (D)

15. When we use n random samples to obtain the OLS regression function the coefficient of determination 2 R is not equal to (A) (B) (C) (D)

17. According to Chebyshev inequality, the probability that the distance between a random variable and the expected value of the random variable is no greater than three times of standard deviation is no less than (A) (B) (C) (D)

18. The probability density function of random variable X is . Var( X) = ? (A) 1 (B) 3 (C) 6 (D)9

19. A normal distribution is with known mean μ and unknown variance θ . If we have n samples , the maximum likelihood estimator of θ is (A) (B) (C)(D)

20. There are two projects as follows:
Project A : If success, the profit is 400,000； if not success : the loss is 40,000.
 Project B : If success, the profit is 500,000； if not success : the loss is 115,000.
 If they have the same success probability and expected profit, the success probability is (A) (B) (C) (D)

II (20%) is the impact of X on Y in the population. If we have 40 random samples of
(i) Find the OLS equation and R² .
 (ii) If we transform the 40 random samples to regress, find the OLS equation

III (20%) Let Y₁ ,Y₂ ,Y₃ ….,Y _n be n uncorrelated random variables with the same mean μ and variance σ²  and denote as the sample average. 

(i) Find E Y( ) and Var Y( ).
(ii) If we define the class of linear estimators of μ as  
 where every a_i  is constant, if G is an unbiased estimator of μ , what restriction on the a_i needed?
 (iii) Find Var (G )
 (iv) Along with (i)-(iii), show that Var G ≥Var  ( ) .