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統計學(A 版)
期中考試
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姓名：
100/4
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( 16%)
(
)1. A sample of k members of a population is said to be a random sample if the members are
chosen in such a way that all possible choices of the k members are equally likely.
(
)2. 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.
(
)3. A scatter diagram is useful in detecting outliers. Outliers are data points that do not
appear to follow the pattern of the other data points.
(
)4. The sample mode is a balancing point called the center of gravity.
(
)5. For some data sets the number of distinct values is too large to utilize. In such cases, we
divide the values into groupings, or class intervals. Generally, 5 to 10 class intervals are
typical.
(
)6. A data set whose histogram has more than two local peaks is said to be bimodal.
(
)7. 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.
(
)8. A couple has two children. Let A denote the event that their older child is a girl, and let B
denote the event that their younger child is a boy. Assuming that all 4 possible
outcomes are equally likely, we know that A and B are independent.
(
)9. If P(B|A) is equal to P(B), then B is independent of A.
(
)10. Consider an experiment whose sample space is S. The probability of sample space S is
less than 1.
(
)11. Let X be a random variable, if Var(X)=4, then SD(3X)=6.
(
)12. 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 ].
(
)13. The standard normal distribution is a normal distribution having variance 0 and mean 1.
(
)14. The trials of a hypergeometric random variable should be independent.
(
)15. 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.
(
)16. All bell-shaped symmetric density curves are normal distribution.
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二、 選擇題(每題 2 分，共 14 分)
(
) 1. Which one is incorrect?
(A) Statistics is the art of learning from data.
(B) The part of statistics concerned with the drawing of conclusions from data is called
inferential statistics.
(C) The part of statistics concerned with the description and summarization of data is
called descriptive statistics.
(D) The total collection of all the elements that we are interested in is called a sample, not
a population.
(
) 2. Which kind of graphs is the right graph?
(A) Bar graph
(B) Frequency polygons
(C) Line graph
(D) Frequency histogram
(
) 3. 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
(
) 4. 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 third quartile is 54.
The data should be symmetric.
The data may be more concentrated in interval [50. 51.5].
) 5. 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 and B > 0.
(C) |A| > |B|.
(D) A  0 and B  0.
) 6. Which one is incorrect?
(A) P( A  B)  P( A | B) P( B)
(B) If A and B are disjoint, then P( A  B)  P( A) P( B)
(C) P( A  B)  P( A)  P( B)  P( A  B)
(D) If A and B are disjoint, then P( A  B)  P( A)  P( B)
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(
) 7. Let X and Y be independent random variables and c is a constant, which one is
incorrect?
(A) Var(X+c) = Var(X) + c2
(B) Var(X+X) = Var(2X)
(C) Var(X+X) = 4Var(X)
(D) Var(X+Y) = Var(X) + Var(Y)
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. (4%) Seventy-five values are arranged in increasing order. How would you determine the
sample (a) 80th percentile (b) 30th percentile of this data set?
3. (2%) Two cards are randomly selected from a deck of 52 playing cards. What is the conditional
probability they are both aces given that they are of different suits?
4. (2%) Let A, B, C be events such that P(A) = 0.2, P(B) = 0.3, P(C) = 0.4. Find the probability
p (( A  B)  C ) if A, B, C are mutually exclusive.
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5. (4%) The following table gives the U.S. per capita consumption (人均消耗量) of whole milk (x)
and of low-fat milk (y) in three different years.
Find the sample correlation coefficient r for the given data.
6. 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)?
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7. (8%) The probability that a fluorescent bulb burns for at least 500 hours is 0.90. Of 8 such bulbs,
find the probability that
(a) All 8 burn for at least 500 hours.
(b) Exactly 7 burn for at least 500 hours.
(c) What is the expected value of the number of bulbs that burn for at least 500 hours?
(d) What is the variance of the number of bulbs that burn for at least 500 hours?
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.
9. (4%) 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?
10. (a) (2%) What is the density function of a normal
distribution?
(b) (2%) Here show a normal distribution
with   2,   2 . Pleas draw another normal
distribution with   2,   0.5 .
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11. (6%) Find (a) P{Z < 1.5}, (b) P{Z ≥ 0.8}, and (c) P{−1.5 < Z < 2.5}.
12. (4%) IQ examination scores for sixth-graders are normally distributed with mean value 100 and
standard deviation 14.2.
(a) What is the probability a randomly chosen sixth-grader has a score greater than 130?
(b) What is the probability a randomly chosen sixth-grader has a score between 90 and 115?
13. (4%) If X is a normal random variable with mean 50 and standard deviation 6, find the
approximate value of x for which (a) P{X > x} = 0.10 and (b) P{X < x} = 0.05.
14. (2%) Find the value of x, to two decimal places, for which P{|Z |< x}=0.99.
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15. (6%) Find (a) z0.07, (b) z0.65, and (c) z0.50.
16. (4%) Suppose the amount of time a light bulb works before burning out is a normal random
variable with mean 400 hours and standard deviation 40 hours. If an individual purchases two
such bulbs, one of which will be used as a spare to replace the other when it burns out, what is
the probability that the total life of the bulbs will exceed 750 hours?
17. (4%) 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?
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