110年 - 110 國立中央大學_碩士班招生考試_工業管理研究所/不分組(一般生):統計學#103859

科目：研究所、轉學考(插大)◆統計學 | 年份：110年 | 選擇題數：0 | 申論題數：12

試卷資訊

所屬科目：研究所、轉學考(插大)◆統計學

選擇題 (0)

申論題 (12)

(a) What is the pdf of X? (5 points)

(b) Determine E[X] for a ＞ 1. (10 points)

(c) What is the moment generating function of X, , where t ≤ O? (10 points) Hint: Use incomplete gamma function:

(d) Let X₁and X₂are independent identically distributed random variables with Pareto distribution with parameter x_m and a defined earlier. Show that the two random variables min{X₁,x₂} and (X₁+ X₂)/min{X₁, X₂} are independent. (10 points)

2.(15 points) Let X and Y be continuous random variables having a joint density. Suppose that Y andφ(X)Y have finite expectation. Show that

(a) Assume the mean filling volumes of two machines are the same. Please state the hypotheses that should be tested in this experiment. (2 points)

(b) Test these hypotheses using a = 0.05. What are your conclusions? (10 points)

(c) Find the p-value for this test. (3 points)

(a) Please use t-test on the equality of mean filling volumes of two machines at a = 0.1. What are your conclusions? (10 points)

(b) Find a 90 percent confidence interval on the difference in mean fill volume for the two machines. (5 points)

(c) Rework the problem/test via the analysis of variance (ANOVA) approach. Prepare an ANOVA table to summarize the results. Show why the conclusion is coincidence to (a). (Hint: the corresponding critical F-value can be calculated from a t-value.) (10 points)

5. Consider the test in Problem 4. If the mean fill volumes of the two machines differ by as much as 5 ml, what is the power of the test used in Problem 4. (give a range based on the provided t-table) (5 points) What sample size would result in a power of at least 0.95 if the actual difference in mean fill volume is 5 ml? (5 points) 651