Download 抽樣方法與抽樣分配

抽樣方法與抽樣分配
1. 請回答下列問題
在個數為 n 的抽樣架構中，每一個體都有一樣的機率被抽驗，這種抽樣方法
稱為__________。
2. Stratified random sampling is a method of selecting a sample in which
(A) the sample is first divided into strata, and then random samples are taken from
each stratum
(B) various strata are selected from the sample
(C) the population is first divided into strata, and then random samples are drawn
from each stratum
(D) None of these alternatives is correct.
3. A statewide sample survey is to be made. First, the state is subdivided into
counties. Seven counties are selected at random and further sampling is
concentrated on these even counties. What type of sampling is this？
(A) Simple random
(B) Nonproportional
(C) Cluster
(D) Stratified
(E) None of the above
4. The number of electronic parts to be examined is 500. The electronic parts are
numbered from 1 to 500. It is decided to use stratified sampling to draw five
sample. The electronic parts are divided into five strata equally, i.e., 1-100,
101-200, 201-300, 301-400, 401-500. If the random numbers chosen are 0.136、
0.585、0.038、0.814、0.594，what are the coded numbers of chosen parts？
5. Statisticians regard a population census as
(A) usually unnecessary
(B) often a waste of resources
(C) often not accessible
(D) all of the above
(E) none of the above
6. Which one of the following doesn’t involve sampling error？
(A) census
(B) stratified sample
(C) sample random sample
(D) cluster sample (E) all of the above
7. Please explain the following terms in detail：
Sample Statistic
8. 解釋下列名詞之意義，並舉一簡例說明之。
Independent identically distribution random variables
9. Let X1 , X 2 ,..., X n be a random sample taken from a population with mean 
n
and variance  2 . Let X   X i n , the sample mean. Which of the following
i 1
statements are correct？
(A) The expected value of X is always equal to population mean  .
(B) The variance of X is always equal to the population variance  2 .
(C) The variance of X is smaller than the population variance  2 , when sampling
from a finite population (有限母體).
(D) The variance of X is larger than the population variance  2 , when sampling
from an infinite population (無限母體).
10. 試證
1
Fu ,v,a 
Fv,u ,1a
11. 試分析常態分配、 t 分配、  2 分配及 F 分配間的關係。
12. As its degrees of freedom tends to infinity, a t distribution converges to
(A) a t distribution
(B) a chi-square distribution
(C) an F distribution
(D) the standard normal distribution
(E) not necessary any of (A) to (D)
13. Which of the following is not a characteristic of the F distribution？
(A) It is a discrete distribution.
(B) It cannot be negative.
(C) It is a based on two sets of degrees of freedom.
(D) All of the above.
14. The distribution of Student’s t has
(A) a mean of zero and a standard deviation of one
(B) a mean of one and a standard deviation of one
(C) a mean of zero and a standard deviation that depends on the sample size
(D) a mean that depends on the sample size and a standard deviation of one
15. 若 x1 , x2 ,..., x9 是隨機抽樣(即 x1 , x2 ,..., x9 獨立)自常態分配，平均數 12、標準差
10(即 x1 , x2 ,..., x9 ~N(12,100))，求下列期望值與機率值：
(1) E  x1  x2  ...  x9   ？
(2) P  x1  x2  x3  x4  68  ？
2
 9
(3) E   xi  x   ？
 i 1



(4) P  x1  x2  x3  x4  88 | x5  10  ？
16. Let X denote the mean of a random sample of size 60 from the distribution
whose probability density function is f  x    3 2 x2 , 1  x  1, please use the
normal approximation to compute the probability that X is between 0.02 and
0.05.
17. The wildlife department has been feeding a special food to rainbow trout
fingerlings in a pond. A sample of the weights of 40 trout revealed that the mean
weight is 402.7 grams and the standard deviation 8.8 grams. What is the
probability that the mean weight for a sample of 40 trout exceeds 405.5 grams？
(A) 0.3783
(B) 0.0228
18. 何謂中央極限定理
(C) 1.0
(D) 0.5
(E) None of the above
19. A theorem that allows us to use the normal probability distribution to approximate
the sampling distribution of sample means and sample proportions whenever the
sample size is large is known as the
(A) Approximation Theorem
(B) Normal Probability Theorem
(C) Central Limit Theorem
(D) Central Normality Theorem
20. A population has a mean of 180 and a standard deviation of 24. A sample of 64
observations will be taken. The probability that the mean from that sample will be
between 183 and 186 is
(A) 0.1359
(B) 0.8185
(C) 0.3413
(D) 0.4772
21. If 10 fair dice are rolled, find the approximate probability that the sum obtained is
between 30 and 40. (Hint: Use central limit theorem to find the answer)
(A) 0.21
(B) 0.32
(C) 0.46
(D) 0.65
(E) 0.84
22. The Central Limit Theorem is important in statistics because
(A) for a large sample size, it says the population is approximately normal.
(B) for any population, it says the sampling distribution of the sample mean is
approximately normal, regardless of the sample size.
(C) for a large sample size, it says the sampling distribution of the sample mean is
approximately normal, regardless of the shape of the population.
(D) for any sized sample, it says the sampling distribution of the sample mean is
approximately normal.
23. The average cost of a one-bedroom apartment in a town is $450 per month. What
is the probability of randomly selecting a sample of 50 one-bedroom apartments in
this town and getting a sample mean of less than $430 if the population standard
deviation is $100.
24. Round-off error has a uniform distribution on [-0.5, +0.5] and round-off errors are
independent. A sum of 50 numbers is calculated where each is of the form XXX.D,
rounded to XXX before adding. What is the probability that the total round-off
error exceeds five？
(A) 0.0071
(B) 0.0081
(C) 0.0091
(D) 0.0101
(E) 0.0111