110年 - [非官方正解]110 國立臺灣大學_碩士班招生考試_財務金融研究所乙組:統計學(I)#105936

科目：研究所、轉學考(插大)◆統計學 | 年份：110年 | 選擇題數：10 | 申論題數：8

試卷資訊

所屬科目：研究所、轉學考(插大)◆統計學

選擇題 (10)

1.Suppose X and Y have a bivariate normal distribution with covariance 1. X has a mean of 10 and standard deviation of 4. Y has a mean of 15 and standard deviation of 6. Z = 2X + 3Y. Which of the following statement is false? (A)The mean of Z, is 65. (B) The variance of Z is 400. (C) The correlation between X and Y is 1/24. (D) Z follows a normal distribution. (E) None of the above choices (a)-(d).
2. Y is distributed N(0,4), and X is distributed N(0,16). Please calculate E(Y2) and E(X3). (A)E(Y2) = 4 and E(X3) = 16 (B) E(Y2) = 2 and E(X3) = 64 (C) E(Y2) = 4 and E(X3) = 0 (D) E(Y2) = 2 and E(X3) = 8 (E) None of the above choices(a)-(d).

3. A random draw (with replacement) of 1000 balls from a pool of several million balls is conducted. Each ball is either blue or red. Let p denote the fraction of blue balls in the population. Let 61ef70ae5b461.jpgdenote the fraction of blue balls in the sample. You want to test the competing hypotheses Ho: p = 0.4 vs. H1: p≠0.4. Suppose that you decide to reject61ef70e55b605.jpg. Note that the sample of 1000 balls is considered as a large sample. What is the type I etror of this test? (A)0.197 (B) 0.261 (C) 0.325 (D) 0.413 (E) 0.522

4.Suppose you estimate a sample OLS regression model using 20 observations (for i = 1,2, ..., 20). The resulted regression equation is: Yi = 2.11 + 3.58Xi +εi. Yi, Xi, and εi denote the dependent variable, independent variable, and the regression residual, respectively. If Xi is equal to 2 for i = 1,2 .... 4； equal to 3 fori = 5,6 ,..., 10； equal to 11 for i = 11, 12； equal to 14 for i =13, 14, ... 20, what is the value of Y? (A)30.75 (B) 28.96 (C) 26.12 (D) It cannot be determined. (E) None of the above choices (a)-(d).

5. Suppose a population regression equation is: Yi=β0 +β1Xi +ui. Yi, Xi, and ui denote the dependent variable, independent variable, and the error term, respectively. i indexes cach individual observation. Which of the following statement is correct? (A) If61ef7216733ed.jpgand the correlation between Xi and ui is zero, the OLS estimator61ef71dc068f0.jpg is unbiased. (B) If61ef7216733ed.jpg and the correlation between Xi； and ui； is zero, the OLS estimator 61ef71dc068f0.jpg is inconsistent. (C) 61ef7216733ed.jpg, the correlation between Xi and ue may not be zero. (D) The assumption that the correlation between Xi and ui is zero cannot be tested empirically. (E)None of the above choices (a)-(d). 
6.In Equation 2, holding all other factors constant, when Age increases by I year, what is the predicted change in Earnings? (A)Increase by 370 dollars. (B) Increase by 1448 dollars. (C) Increasc by 0.37%. (D) Increase by 37%. (E) None of the above choices (a)-(d).
7.Suppose Equation 2 is the correct model. That is, Age is a determinant of a person's carnings. Are older people more likely than younger people to have a college degree in this sample?(A) Older people are more likely to have a college degree. (B) Younger people are more likely to have a college degree.(C) All people are equally likely to have a college degree. (D) We don't have sufficient information to answer this question. (E) None of the above choices (a)-(d).
8. Which of the following is false? (A)If all Gauss-Markov conditions are satisfied, the OLS estimators have the lowest conditional variance among all linear conditionally unbiased estimators. (B) If not all Gauss-Markov conditions are satisfied, the OLS estimators are biased. (C) If the regression error ui is conditionally heteroscedastic, the OLS estimators can still be consistent. (D) If the conditional variance of ui is heteroscedastic and its functional form is known and can be estimated exactly, we may prefer a weighted least square estimator to an OLS estimator because of the efficiency consideration. (E) None of the above choices (a)-(d). 
9. Lot size (Lsize) is measured in square meters. If you change the unit of measurement to square centimeters. What will be the new estimated coefficient of Lsize? (A)61ef75428aef7.jpg(B)61ef755b6c12d.jpg(C) 50 x 102 (D) 50 x 104 (E) None of the above choices(a)-(d).
10. H0: The coefficients of NBR and Age are jointly zero. Ha: The coefficient of NBR and Age are not jointly zero. You H0 using a F-test, with the test statistic = 2.2. Probability(F-statistic ＞ 2.2) = 0.03, Which of the following is correct? (A)You can reject H0 at the 10% level, but not at the 5% level. (B) You can reject H0 at both the 10% and 5% levels, but not at the 1% level. (C) You can reject H0 at the 10%, 5% level, and 1% levels. (D) You cannot reject H0 at the 10%, 5% level, or 1% level. (E) None of the above choices(a)-(d). 

申論題 (8)

1.Suppose that X and Yare two continuous random variables with the joint probability density function /(x.y) - x + 2y for 0 ＜x＜2 and 0 ＜y ＜ 2. The covariance of Xand Y is________.

2.A random variable X has a mean p = 5 and variance σ2 = 36. Based on Chebyshev's inequality, a lower bound on Pr(-7＜X＜ 17)is____________.
3.Suppose that X is a father's height, Y is a mother's height, Z is a child's ultimate height (as an adult), and G is delined as: G = I if the child is a boy and G = -I ifthe child is a girl. Let M = (X+ Y)/2. Suppose that, conditional on known values of G, X, and Y, Z has a continuous uniform distribution with range [M -5 +3G, M + 5 + 3G]. If a girl's father is 4 inches taller than her mother, the probability that she will grow to be taller than her mother is__________.
4.A manufacturer of car tires claims that his tires will last, on the average, 4 years with a standard deviation of 1 year. Given that 5 of these tires have lifetimes of 2.1, 2.2, 3.0, 3.6, 4.1 years, if someone wants to test whether his tires have a standard deviation of1 year, the figure of the appropriate test statistic is equal to__________.

5.You are going to carry out an experiment to determine whether surface finish has an effect on the endurance limit of steel. It is expected that polishing increases the average endurance limit. In such an experiment, you want to detect that polishing fails to have an effect with a probability of 0.99 (i.(E), a = 0.01) and you also want to detect a change in the average endurance limit of 700 units by a probability of at least 0.9. If it is known that the standard deviation of the endurance limit of the steel is 420 units, then the sample size that would be needed to perform this experiment is____________ . Assume that for a = 0.01, Za = 2.5 and forβ -0.10, 2β- 1.5.

6.How large a sample must be taken in order that you are 95 percent certain that61ef8582af7ae.jpg is within 0.5 of the standard deviation of the mean?____________.
7. Ten trials are conducted of a given system with the following results: S, S, F, S, S, S, F, S, S, S ("S" represents success and "IF" represents failure), assuming that the trials are independent of each other. The maximum likelihood estimate of the probability of successful operation is___________.
8.At the beginning of the schoo! year, the mean score of a group of 49 sixth-grade students on an achievement test in reading was 70 with an unbiased sampie variance of 36. At the end of the school year, the mean score on an equivalent form of the same test was 74 with an unbiased sample variance of 25. The correlation between scores made on the initial and final testing was 0.6. To test whether the mean scores on these two exams are significantly different based on the Student t-test, the figure of the test statistic is equal to___________.