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統計學(A 版)
期中考試
學號：
姓名：
99/4
一、 是非題( 15%)
(
)1. The part of statistics concerned with the drawing of conclusions from data is called
descriptive statistics (描述統計). Moreover, the part of statistics concerned with the
description and summarization of data is called inferential statistics (推論統計).
(
)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 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.
(
)4. 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.
(
)5. Sample mean is defined to equal the arithmetic average of the data values. The sample
median is a balancing point called the center of gravity.
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)6. 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.
(
)7. A data set whose histogram has more than two local peaks is said to be bimodal.
(
)8. The sample correlation coefficient r is always between −1 and +1. 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.
(
)9. 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.
(
)10. Consider an experiment whose sample space is S. The probability of sample space S is
less than 1.
(
)11. The probability of the union of disjoint events is equal to the sum of the probabilities of
these events.
(
)12. Let X be a random variable with expected value E[X] and c be a constant, then E[cX] =
cE[X] and E[X + c] = E[X] + c
(
)13. The standard normal distribution is a normal distribution having mean 0 and variance 1.
(
)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. All bell-shaped symmetric density curves are normal distribution.
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二、 簡答題
1. The following data represent the number of days of sick leave (病假) taken by each of 50
workers of a given company over the last 6 weeks:
2, 2, 0, 0, 5, 8, 3, 4, 1, 0, 0, 7, 1, 7, 1, 5, 4, 0, 4, 0, 1, 8, 9, 7, 0,
1, 7, 2, 5, 5, 4, 3, 3, 0, 0, 2, 5, 1, 3, 0, 1, 0, 2, 4, 5, 0, 5, 7, 5, 1
(1) (3%) Please show the frequency table.
(2) (3%) Please draw the bar graph.
(3) (3%) Is it a symmetric bar graph? Why?
2. 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) (3%) Please show its stem-and-leaf plot.
(2) (3%) What is the sample median?
(3) (3%) What are the sample modal values (sample mode)?
3. (4%) Which data value is the sample 90th percentile when the sample size is (a) 8 and (b) 100?
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4. Given two data sets A= {1, 2, 5, 6, 6} and B={ −40, 0, 5, 20, 35}. We know that both the sample
mean and the sample median of the two sets are the same.
(1) (3%) Please calculate the sample variance of data set A.
(2) (3%) Suppose the sample variance of data set B is 792.5. Let C = { 2(−40)+20, 2(0)+20,
2(5)+20, 2(20)+20, 2(35)+20}, please show the sample standard deviation of C.
5. (5%) Given a box plot as follows:
Could you tell us any information about the sample data?
6. (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.
7. (3%) Four fair dice are to be rolled. Find the probability that 6 appears at least once.
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8. 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 if
(1) (3%) A, B, C are independent.
(2) (3%) A, B, C are mutually exclusive.
9. 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?
10. (4%) Suppose that two people are randomly chosen from a group of 4 women and 6 men. What
is the probability that one is a woman and the other a man?
11. (4%) A blood test is 99 percent (0.99) effective in detecting a certain disease when the disease is
present. However, the test also yields a false-positive result for 2 percent (0.02) of the healthy
patients tested. (If a healthy person is tested, then with probability 0.02 the test will say that this
person has the disease.) Suppose 0.5 percent (0.005) of the population has the disease. Find the
conditional probability that a randomly tested individual actually has the disease given that his
or her test result is positive.
12. (6%) Suppose E[X] = μ and SD(X) = σ. Let Y = (X – μ)/σ, show that E[Y] = 0 and Var (Y) = 1.
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13. (4%)
14. (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?
15. The heights of adult males are normally distributed with a mean of 69 inches and a standard
deviation of 2.8 inches. Let X denote the height of a randomly chosen male adult. Find
(a) (3%) P{62 < X < 72}
(b) (3%) P{|X − 69| > 6}
(c) (3%) x if P{X < x} = 0.95
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?
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