25. In country B, men's longevity are investigated and the following results are found: 1. Probability that a man lives at least 70 years is 77%； 2. Probability that a man lives at least 80 years is 48%. What is the conditional probability that a man's age is at least 80 years old given that he bas just celebrated his 70th birthday?
(A)
(B)0.48；
(C)0.77；
(D)1；
(E)None of the above is correct. e, a and c.

1. Consider n samples independently drawn from a distribution with mean O and variance 1. What is the minimum value of n that will make variance of the sample mean no greater than 0.01? (A) 30；(B)4；(C)100；(D)n does not matter here；(E)None of the above is correct.

2. A 95% confidence interval of a population mean is . Which of the following statement is the most correct one? (A) If random samples were drawn again and again, with computed each time, the population mean would lie in the interval 95% of the samples. (B)If = 0.5, the population mean would lie in the interval [0.1,0.5] with 95% probability. (C) If = 0.55, the population mean might still lie outside the interval [0.05,0.55)(D) If= 0.56, the population mean would ie in the interval [0.03, 0.56] greater than 95% probability. (E) a and c.

3. 1f E[XIY] = E[X], then (A) X and Y are mutually independent；(B)X and Y are uncotrelated；(C)E [X2|Y] = E [x2]；(D) Var (X|Y) = Var(X)；(E)E(X]=0.

4. Consider the following hypothesis test about population mean: H0 :μ = 3, H1 : μ ＞ 3. Which of the following results gives most supportive evidence to reject the hypothesis? (A) Sample mean equals to 5 and standard error equals to 1. (B)Sample mean equals to 5 and standard error equals to 2. (C) Sarple mean equals to 3.5 and standard error equals to 1. (D) Sample mean equals to 2 and standard error equals to 1.(E) Sarple mean equals to 2 and standard error equals to 0.5.

5. Suppose we want to test H0 :μ= μo against H1:μ ≠ μ0: (A) The power function of the test is a function of tu and describes the probability of rejecting H0. (B)The maximun probability of comnitting a Type I error is called the power of the test.(C)  The minimum probability of committing a Type II error is called tbe size of the test.(D) When the power function is evaluated atμo, the value equais to the probability of committing a Type II error.(E)None of the above is correct.

6. Consider two continuous random variables (X,Y) with X [0,∞] and Y  [1,5]. The joint density function of (X,Y) isWhat is the conditional density function of X given Y = 1 if X  [0,∞]? (A) (B)；(C)exp(-x)；(D)；(E) .

7. Consider an experiment having two outcomes: success with probability p = 0.12 and failure with probability 1 - p = 0.88. Suppose we independently conduct the experiment n times and estimate p with  What is the minimum value of n that will make standard deviation of no greater than 0.01? (A) 30；(B)1056；(C)33；(D)n does not matter here,(E)None of the above is correct.

8. A population is composed of 5 numnbers: 2, 4, 6, 8, 10. Let u and a denote mean and standard deviation of the population, and denote mean and standard deviation of the sample mean from the population. Which of the following statement is the most correct one? (A) μ = 6 ，σ = 3 16(B)(C) = 283；(D)μ= 6, σ  = 2.83(E).=μ， σ = 3.16.

9. If we calculate sample mean from the population in question 8, with sample size equal to 2. Let denote the theoretical density and cumulative distributions of the sarple mean, respectively. Which of the following statement is the most correct one? (A)(B)(C) (D)(E)

10. If we calculate a sample mean from the population in question 8, with sarple size equal to 2, Letdenote mean and standard deviation of the sample mean. Which of the following statement is the most correct one? (A)(B)(C)(D)=5.5, = 1.89； (E)

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