課程名稱︰統計學上 課程性質︰必修 課程教師︰關秉宗 開課學院：生農學院 開課系所︰森林環資系 考試日期（年月日）︰2012/01/12 考試時限（分鐘）：180 mins 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Statistics Fall 2011 Final Examine January 12, 2012 I. Let X~N(5,16), please calculate the following probabilities (10%) 1. P(X <= 5.0) 2. P(X <= 8.8) 3. P(X <= 11.0) 4. P(X <= 14) 5. P(X <= 17.0) 6. P(X <= -1.0) 7. P(X >= 0.0) 8. P(X >= 15) 9. P(X >= -7.0) 10. P(-1.0 <= X <= 11.0) II. Let X~t(19), please find the constant c for each of the following questions (5%) 1. P(|X| >= c) = 0.05 2. P(|X| >= c) = 0.01 3. P(X >= c) = 0.025 4. P(|X| >= c) = 0.95 5. P(X <= c) = 0.90 _ III. Let X~(μ,σ^2), X and S^2 be the sample mean and variance of a random _ sample of size 16 from X, with μ unknown. Our sample yielded X = 27.38 and s = 2.26. Please find a 95% confidence interval for μ when (1) σ^2 is known and σ^2 = 4 (5%) (2) When σ^2 is unknown (5%) (3) Please alse find a 95% for σ^2 when σ^2 is unknown (5%) IV. (1). Let S^2 be the sample variance of a random sample of size 6 from N~(μ,12). Please find P(2.75 < S^2 < 26.57). (5%) _ (2). Let X and S^2 be the sample mean and variance of a random sample of _ size 16 from N(3,100). Please compute P(3 < X < 7.112, 41.75 < S^2 < 183.25) (5%) _ _ (3). Let X and Y be the sample means of two independent random samples of _ _ size 25 from X~N(0,16) and Y~N(1,9). Please compute P(X > Y). (5%) V. Let the life time of light bulbs of brand X be N~(μx,900), and the life of brand Y light bulbs be N~(μy,784). Now we randomly and independently selected 64 light bulbs from X and yeilded a mean of 950 hours, and randomly and independently selected 49 light bulbs from Y and yielded a mean of 980 hours. (1) Please test the hypothesis that the life times of the two brands are equal versus that the two brands have different life times. Please state your Ho and Ha, Let the type I error rate of the test be 0.05. (10%) (2) Please find a 95% C.I. for (μx-μy). (5%) (3) Does the 95% confidence interval you obtained in (b) supports your conclusion in (a)? Why? (5%) VI. Our Experiment Forest wants to conduct an experiment to determine whether a new type of fertilizer will improve seedling height growth of Taiwania. Let the seedling height without fertilization be X~N(35,100). To test this, we now randomly select a number of seedlings from the fertilization plot and measured their height. You can assume that fertilization will not change the variance. (1) Please also state the appropriate Ho and Ha. (2) If the sample mean we obtained is 38, and we want to claim that fertilization is indeed effective in promoting height growth, what will be the minimum sample size if we set α = 0.01? (10%) VII. Let X and Y be two independently distributed normal distributions; X~N(μx,σx^2), and Y~N(μy,σy^2). (註：σx^2代表σx的平方) (1) Let U = (aX+b) + (cY+d), then (A) What is the distribution of U; (B) E(U) = ? (C) Var(U) = ? (5%) _ (2) Let X be the sample mean and Sx^2 be the sample variance of a sample _ of size nx from X, and Y be the sample mean and Sy^2 be the sample variance of a sample of size ny from Y. _ _ (X - μx) (Y - μy) (a) What are the distributions of ──── and ──── ? (5%) Sx/√nx Sy/√ny (註：√nx為nx的平方根) (b) What are the distributions for the squared terms for the above two items? (5%) _ _ (c) Let V = (X - Y); What is the distribution of V? E(V) = ? and Var(V) = ? (5%) (Sx^2/σx^2) (d) Let W = ──────, then what is the distribution of W? (5%) (Sy^2/σy^2) --

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