會計系統計學期中作業
1.
繳交截止日期: 2023/10/27(五)23:59
All highway bridges in the United States are inspected periodically for structural
deficiency by the Federal Highway Administration (FHWA). Data from the
FHWA inspections are compiled into the National Bridge Inventory (NBI).
Several of the nearly 100 variables maintained by the NBI are listed below.
Classify each variable as quantitative or qualitative.
(a) Length of maximum span (feet)
(b) Number of vehicle lanes
(c) Toll bridge (yes or no)
(d) Average daily traffic
(e) Condition of deck (good, fair, or poor)
(f) Bypass or detour length (miles)
(g) Route type (interstate, U.S., state, country, or city)
2.
A researcher conducts a mileage economy test involving 80 cars. The frequency
distribution describing average miles per gallon (mpg) appears in the following
table.
Average(mpg)
Frequency
15 up to 20
15
20 up to 25
30
25 up to 30
15
30 up to 35
10
35 up to 40
7
40 up to 45
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a. Construct the relative frequency, cumulative frequency, and cumulative
relative frequency distributions.
b. How many of the cars got less than 30 mpg? What percentage of the cars got
at least 20 but less than 25 mpg? What percentage of the cars got less than 35
mpg? What percent got 35 mpg or more?
c. Graph the relative frequency histogram. Comment on the shape of the
distribution.
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3.
A categorical variable has three categories, with the following frequencies of
occurrence:
Category
Frequency
A
13
B
28
C
9
(a) Compute the percentage of values in each category.
(b) What conclusions can you reach concerning the categories?
4.
Monthly stock prices (in $) for two competing firms are as follows.
Month
Firm A
Firm B
January
28
21
February
31
24
March
32
24
April
35
27
May
34
25
June
28
20
a. Calculate the sample mean, the sample variance, and the sample standard
deviation for each firm's stock price.
b. Which firm had the higher average stock price over the time period?
c. Which firm's stock price had greater variability as measured by the standard
5.
The manager at a water park constructed the following frequency
distribution to summarize attendance for 60 days in July and August.
Attendance
Frequency
1,000 up to 1,250
5
1,250 up to 1,500
6
1,500 up to 1,750
10
1,750 up to 2,000
20
2,000 up to 2,250
15
2,250 up to 2,500
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a. Calculate the mean attendance.
b. Calculate the variance and the standard deviation.
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6.
The National Sporting Goods Association (NSGA) conducted a survey of the
ages of individuals that purchased skateboarding footwear. The ages of this
survey are summarized in the following percent frequency distribution.
Suppose the survey was based on a sample of 200 individuals. Calculate the
mean and the standard deviation of the age of individuals that purchased
skateboarding shoes. Use 10 as the midpoint of the first class.
Age of User
7.
Percent
Under 14 years old
35
14 to 17
41
18 to 24
15
25 to 34
4
35 to 44
4
45 to 64
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The following set of data is from a sample of n = 5:
(a) Calculate the mean, median, and mode.
7 -5 -8 7 9
(b) Calculate the range, variance, standard deviation, and coefficient of variation.
(c) Calculate the Z scores. Are there any outliers?
(d) Describe the shape of the data set.
8.
假設股價每天的上漲或下跌均為獨立事件且具有相同分配的隨機變數。今
有一家公司的股價，其上漲的機率為 p。請問該公司股價連續在 5 個交易
日都上漲的機率為何？而該公司股價先在兩個交易日之內均下跌，但在隨
後的 5 個交易日均上漲的機率又為何？
9.
參加校慶活動的學生中，大一、大二、大三、大四的學生各佔四分之一，
其中大一學生中的百分之三十五，大二學生中的百分之二十五，大三學生
中的百分之二十，大四學生中的百分之十五為女生。如今隨機抽出一名學
生參加校長餐會。如果抽出的為女生，則其為大一學生的機率為何？其為
大三學生的機率為何？
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10. The probability that an Avon salesperson sells beauty products to a prospective
customer on the first visit to the customer is 0.4. If the salesperson fails to make
the sale on the first visit, the probability that the sale will be made on the second
visit is 0.65. The salesperson never visits a prospective customer more than
twice. What is the probability that the salesperson will make a sale to a particular
customer?
11. According to the CGMA Economic Index, which measures executive sentiment
across the world, 18% of all respondents expressed optimism about the global
economy (www.aicpa.org, March 29, 2012). Moreover, 22% of the respondents
from the United States and 9% from Asia felt optimistic about the global
economy.
a. What is the probability that an Asian respondent is not optimistic about the
global economy?
b. If 28% of all respondents are from the United States, what is the probability
that a respondent is from the United States and is optimistic about the global
economy?
c. Suppose 22% of all respondents are from Asia. If a respondent feels optimistic
about the global economy, what is the probability that the respondent is from
Asia?
12. There are four people being considered for the position of chief executive officer
of Dalton Enterprises. Three of the applicants are over 60 years of age. Two are
female, of which only one is over 60.
a. What is the probability that a candidate is over 60 and female?
b. Given that the candidate is male, what is the probability he is less than 60?
c. Given that the person is over 60, what is the probability the person is female?
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13. Consider the following probability contingency table:
B1
B2
A1
0.25
0.35
A2
0.3
0.1
Find
(a) P(B1| A2)
(b) P(A1| B2)
(c) Whether events A and B are independent using the conditional probability
found in (a).
(d) Whether events A and B are independent using multiplication rule.
14. Given P(A) = 0.2, P(B) = 0.75 and P(A and B) = 0.15. Are event A and B
independent?
15. 在一場有 500 題是非題（每題 0.2 分）的考試中，學生若以拋擲一枚均勻
銅板的結果來決定答案。請問該學生得分的機率分配，平均數與變異數為
何？該學生得 60 分以上的機率為何？該學生得 60 分以上的機率為何？得
20 分以下的機率又為和？
16. The U.S. government has devoted considerable funding to missile defense
research over the past 20 years. The latest development is the Space-Based
Infrared System (SBIRS), which uses satellite imagery to detect and track
missiles (Chance, Summer 2005). The probability that an intruding object (e.g., a
missile) will be detected on a flight track by SBIRS is .8. Consider a sample of
20 simulated tracks, each with an intruding object. Let x equal the number of
these tracks where SBIRS detects the object.
(a) Demonstrate that x is (approximately) a binomial random variable.
(b) Give the values of p and n for the binomial distribution.
(c) Find P(x = 15), the probability that SBIRS will detect the object on exactly
15 tracks.
(d) Find P(x ≥ 15), the probability that SBIRS will detect the object on at least 15
tracks.
(e) Find E(x) and interpret the result.
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17. Many U.S. households still do not have Internet access. Suppose 20 out of 80
households in a small southern town do not have Internet access. A company that
provides high-speed Internet has recently entered the market. As part of the
marketing campaign, the company decides to randomly select ten households
and offer them free laptops along with a brochure that describes their services.
The aim is to build goodwill and, with a free laptop, tempt nonusers into getting
Internet access.
a. What is the probability that six laptop recipients do not have Internet access?
b. What is the probability that at least five laptop recipients do not have Internet
access?
c. What is the probability that two or fewer laptop recipients do not have Internet
access?
d. What is the expected number of laptop recipients who do not have Internet
access?
18. The Orange County Register, as part of its Sunday health supplement, reported
that 64% of America men over the age of 18 consider nutrition a top priority in
their lives. Suppose we select a sample of 60 men. What is the likelihood that:
a. 32 or more consider nutrition important?
b. 44 or more consider nutrition important?
c. More than 32 but fewer than 43 consider nutrition important?
d. Exactly 44 consider diet important?
19. 假設某超級市場中，顧客到收銀台付款的間隔時間為指數分配，而預期間
隔時間為 5 分鐘。現有一位顧客已來到收銀櫃台，請問第二位顧客在 2 分
鐘以後才到達的機率為何？請分別利用指數分配和波式分配來計算，並說
明兩者之間的關係。
20. A large technology firm receives an average of 12 new job applications every 10
days for positions that are not even advertised. Suppose the number of job
applications received follows a Poisson distribution.
a. What is the average number of days between successive job applications?
b. What is the probability that the next job application is received within a day?
c. What is the probability that the next job application is received between the
next 1 and 2 days?
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21. A study of the checkout lines at the Safeway Supermarket in the South Strand
area revealed that between 4 and 7 p.m. on weekdays there is an average of four
customers waiting in line. What is the probability that you visit Safeway today
during this period and find:
a. No customers are waiting?
b. Four customers are waiting?
c. Four or fewer are waiting?
d. Four or more are waiting?
22. The random variable x is best described by a uniform probability distribution
with c = 100 (lower bound) and d = 200 (upper bound). Find the probability that
x assumes a value
(a) Within 10 unit from μ.
(b) Larger than 130.
(c) Between 90 and 160.
23. The monthly rate of return of a stock is a measure investors frequently use for
evaluating the behavior of a stock over time. A stock’s monthly rate of return
generally reflects the amount of money an investor makes (or loses if the return
is negative) for every dollar invested in the stock in a given month. In his classic
text, Foundations of Finance (1976), Eugene Fama demonstrated that the
probability distribution for the monthly rate of return of a stock can be
approximated by a normal probability distribution. Suppose the monthly rates of
return to stock ABC are normally distributed with mean .05 and standard
deviation .03, and the monthly rates of return to stock XYZ are normally
distributed with mean .07 and standard deviation .05. Assume that you have $100
invested in each stock.
(a) Over the long run, which stock will yield the higher average monthly rate of
return? Why?
(b) Suppose you plan to hold each stock for only one month. What is the
expected value of each investment at the end of one month?
(c) Which stock offers greater protection against incurring a loss on your
investment next month? Why?
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