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統計學(A 版)
期中考試
學號：
姓名：
101/4
注意：考試請勿作弊，違規者學期成績以零分計算並送本系學生獎懲委員會處理。
一、 是非題( 18%)
(
)1. Statistics (統計學) is the art of learning from data. The part of statistics concerned with
the drawing of conclusions from data is called inferential statistics (推論統計).
Moreover, the part of statistics concerned with the description and summarization of
data is called descriptive statistics (描述統計).
(
)2. The total collection of all the elements that we are interested in is called a population. A
subgroup of the population that will be studied in detail is called a sample.
(
)3. A set of data is said to be symmetric about the value x0 if c that the frequencies of the
values x0 − c and x0 + c are the same.
(
)4. Sample median is defined to equal the arithmetic average of the data values. The sample
mode is a balancing point called the center of gravity.
(
)5. The sample 100p percentile is that data value having the property that at least 100p
percent of the data are greater than or equal to it.
(
)6. If r is the sample correlation coefficient for the data xi, yi, i = 1,...,n, then for any
constants a, b, c, d, r is also the sample correlation coefficient for the data a+bxi, c+dyi ,
i = 1,...,n.
(
)7. If P(B|A) is equal to P(A), then B is independent of A.
(
)8. For any two events A and B, P(A) = P(A|B) P(B) + P(A|Bc) P(Bc).
(
)9. The probability of the union of disjoint events is equal to the sum of the probabilities of
these events.
(
)10. Let X be a random variable, if Var(X) = 4, then SD(4X) = 8 and Var(X+X) = 8.
(
)11. Let X and Y be random variables with expected values E[X] and E[Y], respectively, then
E[X+Y] = E[X]+ E[Y ] and E[X Y] = E[X] E[Y ].
(
)12. The standard normal distribution is a normal distribution having mean 0 and variance 1.
(
)13. The trials of a hypergeometric random variable are dependent.
(
)14. Probability density function is a curve which associated with a continuous random
variable. The probability that the random variable is between two points is equal to the
area under the curve between these points.
(
)15. The sign of correlation coefficient gives the direction of the relation. It is positive when
the linear relation is such that smaller y values tend to go with smaller x values and
larger y values with larger x values.
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(
)16. The central limit theorem states that the sum of a large number of independent random
variables is approximately normally distributed.
(
)17. If the underlying population distribution is normal, then the sample mean will also be
normal, no matter what the sample size is.
(
)18. The expected value of a chi-squared random variable
= 1, where Zi is used to
represent a standard normal random variable.
二、 選擇題(每題 2 分，共 10 分)
(
) 1. Which kind of graphs is the right graph?
(A) Bar graph
(B) Frequency polygons
(C) Line graph
(D) Frequency histogram
(
) 2. Which kind of data representation methods
is often used to plot relative frequencies when the data are nonnumeric?
(A) Pie chart
(B) Scatter diagram
(C) Frequency histogram
(D) Stem-and-leaf plot
(
) 3. Given a box plot as follows, which one is incorrect?
(A)
(B)
(C)
(D)
(
(
The data go from a low of 47 to a high of 60.
The value of the sample mean is 51.5.
The value of the first quartile is 50.
The data may be more concentrated in interval [50. 51.5].
) 4. Given two data sets shown as follows. If r is the sample correlation coefficient of the
data set, which statement is incorrect?
(A) The sample correlation coefficient r is always between −1 and +1.
(B) A > 0, B < 0 and |A| > |B|.
(C) The sign of r is positive when the linear relation is such that smaller y values tend
to go with smaller x values and larger y values with larger x values and.
(D) A value of |r| of about 0 means that the linear relation is relatively strong.
) 5. Let X and Y be independent random variables and c is a constant, which one is
incorrect?
(A) Var(X+c) = Var(X)
(B) Var(c X) = c2Var(X)
(C) SD(X+X) = 4SD(X)
(D) SD(X+Y) = SD(X) + SD(Y)
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三、 簡答題
1. The following data represent the proportion of public elementary school students that are
classified as minority (少數民族) in each of 18 cities.
55.2, 47.8, 44.6, 64.2, 61.4, 36.6, 28.2, 57.4, 41.3, 44.6, 55.2, 39.6, 40.9, 52.2, 63.3, 34.5,
30.8, 45.3
(1) (2%) Please show its stem-and-leaf plot.
(2) (2%) What is the sample median?
2. (2%) Seventy-five values are arranged in increasing order. How would you determine the
sample 80th percentile of this data set?
3. (2%) Let A, B, C be events such that P(A) = 0.2, P(B) = 0.3, P(C) = 0.4. Find the probability
that all of the events A, B, C occur (P(A,B,C)) if A, B, C are independent.
4. (6%) The probability that a fluorescent bulb burns for at least 500 hours is 0.90. Of 8 such bulbs,
find the probability that
(a) Exactly 7 burn for at least 500 hours.
(b) What is the expected value of the number of bulbs that burn for at least 500 hours?
(c) What is the variance of the number of bulbs that burn for at least 500 hours?
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5. (3%) Suppose E[X] = μ and SD(X) = σ. Let Y = (X – μ)/σ, show that E[Y] = 0 and Var (Y) = 1.
6. (2%) Here show a normal distribution
with   2,   2 . Pleas draw another normal
distribution with   2,   0.5 .
7. Suppose that when two dice are rolled, each of the 36 possible outcomes is equally likely.
(1) (2%) Find the probability that the sum of the dice is 6.
(2) (2%) Suppose that the first die lands on 4, what is the resulting probability that the sum of
the dice is 10?
8.
(4%)The return from a certain investment (in units of $1000) is a random variable X with
probability distribution P{X = − 1} = 0.7, P{X = 4} = 0.2, and P{X = 8} = 0.1 .
Find Var(X), the variance of the return.
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9. (4%)
10. An insurance company believes that people can be divided into two classes — those who are
prone (易於) to have accidents and those who are not. The data indicate that an accident-prone
person will have an accident in a 1-year period with probability 0.1; the probability for all others
is 0.05. Suppose that the probability is 0.2 that a new policyholder is accident-prone.
(a) (2%) What is the probability that a new policyholder(投保人) will have an accident in the
first year?
(b) (2%) If a new policyholder has an accident in the first year, what is the probability that he or
she is accident-prone(特別易出事故的)?
11. (2%) If X is the total number of successes that occur in n trials, then X is said to be a binomial
random variable with parameters n and p. Suppose we are interested in the probability that 3
independent trials, each of which is a success with probability p, will result in a total of 2
successes. What is the probability of a total of 2 successes in the 3 trials?
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12. (4%) IQ examination scores for sixth-graders are normally distributed with mean value 100 and
standard deviation 14.2.
(a) The top 1 percent of all scores is in what range?
(b) What is the probability a randomly chosen sixth-grader has a score between 90 and 115?
13. (2%) Data from the U.S. Department of Agriculture (農業部) indicate that the annual amount of
apples eaten by a randomly chosen woman is normally distributed with a mean of 19.9 pounds
and a standard deviation of 3.2 pounds, whereas the amount eaten by a randomly chosen man is
normally distributed with a mean of 20.7 pounds and a standard deviation of 3.4 pounds.
Suppose a man and a woman are randomly chosen. What is the probability that the woman ate a
greater amount of apples than the man?
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14. (6%) (a) If X is a normal random variable with mean 50 and standard deviation 6, find the
approximate value of x for which P{X > x} = 0.10.
(b) Find the value of x, to two decimal places, for which P{|Z |< x}=0.99.
(c) Find z0.65.
15. (2%) The length of time that a new hair dryer functions before breaking down is normally
distributed with mean 40 months and standard deviation 8 months. The manufacturer is thinking
of guaranteeing each dryer for 3 years. What proportion of dryers will not meet this guarantee?
16. (6%) If you place a $1 bet on a number of a roulette wheel, then either you win $35, with
probability 1/38, or you lose $1, with probability 37/38. Let X denote your gain on a bet of this
type.
(a) Find E[X] and SD(X).
Suppose you continually place bets of the preceding type. Show that
(b) The probability that you will be winning after 1000 bets is approximately 0.39.
(c) The probability that you will be winning after 100,000 bets is approximately 0.002.
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