Download 一、 Multiple Choice and True/False (30%, 3% each):

八十九學年度第二學期
統計學(二)
Mid-term Exam (C 卷)
17/05/04
一、Multiple Choice and True/False (9%, 3% each):
1. If the mean and variance for the population are 900 and 9 respectively, what are the mean and
variance of the sample mean if the sample size is 16?
a. 900, 144
b. 900, 0.5625
c. 900, 1296
d. 900, 9
2. A random sample of size 100 is drawn from a binomial distribution with mean 32 and variance
16. What is the distribution of ( X  32) /( 4 / 10) ?
a.
b.
c.
d.
Normal with mean 32 and variance 1.6
Normal with mean 0 and variance 1
Binomial with n = 64 and p = 0.5
None of the above
3. X is distributed with  = 10 and 2 = 4. Y is distributed with  = 50 and 2 = 100. Which
one has higher coefficient of variation?
a. X
b. Y
c. the same
d. couldn’t tell
二、Short Answer: (21%)
1. (6%) 若某路口每月（以 30 天計）發生車禍之次數為波以松分布，平均次數為 6 次，則該
路口發生車禍之間隔時間為________分布，平均間隔時間為_________天。
2. (3%) List the formula for Chebyshev’s Inequality:
3. (3%) If sample mean = 100, sample standard deviation = 10, and sample size = 16, what is the
stand error of the sample mean?_____________________________________________
4. (6%) 某電子公司生產一產品，該產品之電阻值呈常態分布( = 100,  = 10)
a. 試問該產品一批(10,000 個)，約有多少個超過 120？__________________
b. 隨機抽樣 25 個，其平均電阻值小於 96之機率為多少？_______________
5. (3%) A random sample of 36 observations has been drawn from a normal distribution with mean
50 and standard deviation 12. The probability that the sample mean is in the interval [47,
53] is __________.
1
班級：
學號：
姓名：
三、Open Question: (70%)
1. (15%) Assume that one has 2 independent random samples of sizes n1, n2 from the same


n X  n2 X 2
X  X2
population. Show that  wt  1 1
is a better estimator for  than   1
.
n1  n2
2
2. (15%) 請說明: a. 中央極限定理; b. 大數法則; c. 常態分布之標準化。
3. (10%) Let X be a random variable with the following probability distribution:
e  x
f x  
, x  1,2,....
x!
Find the maximum likelihood estimator of , based on a random sample of size n.
2
八十九學年度第二學期
統計學(二)
Mid-term Exam (C 卷)
17/05/04
4. (15%) The following data are the temperatures of effluent at discharge from a sewage treatment
facility on consecutive days:
43 47 51 48 52 50 46 49 45 52 46 51
44 49 46 51 49 45 44 50 48 50 49 50
a. Calculate the sample mean and sample variance.
b. Construct a stem-and-leaf plot of the data.
c. Construct a box plot of the data and comment on the information in this display.
 X
n
5. (15%) Show that
i 1
i
X

2
n
is a biased estimator of 2.
this estimator.
3
In addition, what is the bias of