Sample final exam: statistics and probability analysis môn Xác suất thống kê| Trường Đại học Ngoại Thương

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Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu TEST 1 Time: 60 minutes
Question 1. (2 marks) In 2011, there were 1901 fatalities recorded on Britain’s roads, 60 of
which were for children (Department of Transport, 2012). Correspondingly, serious injuries
total ed 23 122 of which 20 770 were for adults.
a. (1 mark) What is the probability of a serious injury given the victim was a child?
b. (1 mark) What is the probability that the victim was an adult given a fatality occurred? Question 2. (2 marks)
The accounting department at Weston Materials Inc., a national manufacturer of unat ached
garages, reports that it takes two construction workers a mean of 32 hours and a standard
deviation of 2 hours to erect the Red Barn model. Assume the assembly times fol ow the normal distribution.
a. (1 mark) What percent of the garages take between 29 hours and 34 hours to erect?
b. (1 mark) Of the garages, 5 percent take how many hours or more to erect? Question 3. (1 mark)
Using historical records, the personnel manager of a plant has determined the probability
distribution of X, the number of employees absent per day. It is X 0 1 2 3 4 5 6 7 P(X) 0,005 0,025 0,31 0,34 0,22 0,08 0,019 0,001
Calculate the mean and the standard deviation.
Question 4. (3 marks) Rut er Nursery Company packages its pine bark mulch in 50 pound
bags. From a long history, the production department reports that the distribution of the bag
weights fol ows the normal distribution and the standard deviation of this process is 3
pounds per bag. At the end of each day, Jeff Rut er, the production manager, weighs 10 bags
and computes the mean weight of the sample. Below are the weights of 10 bags from today’s production.
45,6 47,7 47,6 46,3 46,2 47,4 49,2 55,8 47,5 48,5 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu
a. (1 mark) Calculate the sample mean and standard deviation.
b. (1 mark) Contruct a 95 percent confidence interval for the population mean.
c. (1 mark) Can Mr. Rut er conclude that the mean weight of the bags is less than 50 pounds
? Use the 0.01 significance level.
Question 5. (2 marks) Research in the gaming industry showed that 10 percent of al slot
machines in the United States stop working each year. Short’s Game Arcade has 60 slot
machines and only 3 failed last year. Use the hypothesis-testing procedure at the 0.05
significance level to test whether this data contradicts the research report.
a. (1 mark) State the nul and the alternative hypotheses.
b. (1 mark) What is the conclusion of the hypothesis testing? 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu TEST 2 Time: 60 minutes
Question 1. (2 marks) Data evidence shows that last year 25% of the stocks in a stock
exchange performed wel , 25% poorly, and the remaining 50% performed on average.
Moreover, 40% of those that performed wel were rated a “good buy” by a stock analyst, as
were 20% of those that performed on average, and 10% of those that performed poorly.
a. (1 mark) What is the probability that a stock rated were rated a “good buy"?
b. (1 mark) What is the probability that a stock rated a “good buy" by the stock analyst wil perform wel this year?
Question 2. (2 marks) A recent report in USA Today indicated a typical family of four
spends $490 per month on food. Assume the distribution of food expenditures for a family
of four fol ows the normal distribution, with a mean of $490 and a standard deviation of $90.
a. (1 mark) What percent of the families spend more than $30 but less than $490 per month on food?
b. (1 mark) What percent of the families spend less than $430 per month on food?
Question 3. (1 mark) The fol owing density function describes the random variable X.  x  x   ; 0 5 25  f x  1  0 ; 5x 10   x    25 0  ; 0;x10  
Compute the probability that X is less than 7.
Question 4. (1 mark) A corporation is considering a new issue of convertible bonds.
Management believes that the offer terms wil be found at ractive by 20% of al its current
stockholders. Suppose that this belief is correct. A random sample of 130 current
stockholders is taken. What is the probability that the sample proportion is between 0.18 and 0.22? 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu
Question 5. (2 marks) A survey of 750 university students found they were paying on
average $108 per week in accommodation costs. Assume the population standard deviation
for weekly accommodation costs is $22.
a. (1 mark) Construct a 90 per cent confidence interval estimate of the population mean.
b. (1 mark) Construct a 95 per cent confidence interval estimate of the population mean.
Question 6. (2 marks) After many years of teaching, a statistics professor computed the
variance of the marks on her final exam and found it to be 2  = 250. She recently made
changes to the way in which the final exam is marked and wondered whether this would
result in a reduction in the variance. A random sample of this year’s final exam marks are listed here. 57 92 99 73 62 64 75 70 88 60
a. (1 mark) Calculate the sample mean and variance.
b. (1 mark) Can the professor infer at the 10% significance level that the variance has decreased? 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu TEST 3 Time: 60 minutes
Question 1. (2 marks) A union’s executive conducted a survey of its members to determine
what the membership felt were the important issues to be resolved during upcoming
negotiations with management. The results indicate that 74% of members felt that job
security was an important issue, whereas 65% identified pension benefits as an important
issue. Of those who felt that pension benefits were important, 60% also felt that job security
was an important issue. One member is selected at random.
a. (1 mark) What is the probability that he or she felt that both job security and pension benefits were important?
b. (1 mark) What is the probability that the member felt that at least one of these two issues was important?
Question 2. (2 marks) According to a Gal up pol 27% of American adults have confidence
in banks. Suppose that you interview 5 Americans adults at random.
a. (1 mark) What is the probability that 2 or fewer have confidence in banks?
b. (1 mark) What is the probability that no one had confidence in banks?
Question 3. (1 mark) A random variable has the fol owing density function.  xx   f x  1  ;0 2  2 x   0   ; 0; 2
Calculate the mean and variance of the random variable.
Question 4. (2 marks) The weights of a random sample of cereal boxes that are supposed
to weigh 1 pound are listed here.
1,05 1,03 0,98 1,00 0,99 0,97 1,01 0,96
a. (1 mark) Calculate the sample mean and variance.
b. (1 mark) Assume the normality, estimate the variance of the entire population of cereal
box weights with 90% confidence. 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu
Question 5. (1 mark) Fuel consumption tests are conducted for a particular model of car. If
a 98 per cent confidence interval with a margin of error of 0.2 litres per 100km is desired,
how many cars should be used in the test? Assume that preliminary tests indicate the
standard deviation is 0.5 litres per 100km.
Question 6. (2 marks) Has the recent drop in airplane passengers resulted in bet er on-time
performance? Before the recent downturn one airline bragged that 92% of its flights were on
time. A random sample of 165 flights completed this year reveals that 153 were on time.
Can we conclude at the 5% significance level that the airline’s on-time performance has improved?
a. (1 mark) State the nul and the alternative hypotheses.
b. (1 mark) What is the conclusion of the hypothesis testing? 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu TEST 4 Time: 60 minutes
Question 1. (2 marks) In a large corporation, 80% of the employees are men and 20% are
women. The highest levels of education obtained by the employees are graduate training for
10% of the men, undergraduate training for 30% of the men, and high school training for
60% of the men. The highest levels of education obtained are also graduate training for 15%
of the women, undergraduate training for 40% of the women, and high school training for 45% of the women.
a. (1 mark) What is the probability that a randomly chosen employee wil be a man with only a high school education?
b. (1 mark) What is the probability that a randomly chosen employee who has graduate training is a man?
Question 2. (2 marks) Andrew Whit aker, computer center manager, reports that his
computer system experienced three component failures during the past 100 days.
a. (1 mark) What is the probability of no failures in a given day?
b. (1 mark) What is the probability of at least two failures in a 3-day period?
Question 3. (1 mark) Forest Green Brown, Inc., produces bags of cypress mulch. The
weight in pounds per bag varies, as indicated in the accompanying table. Weight 44 45 46 47 48 49 50 Proportion 0,04 0,13 0,21 0,29 0,2 0,1 0,03
The cost (in cents) of producing a bag of mulch is 75 + 2X, where X is the number of
pounds per bag. The revenue from sel ing the bag, regardless of weight, is $2.50. If profit is
defined as the difference between revenue and cost, find the mean and standard deviation of profit per bag.
Question 4. (1 mark) It is believed that first-year salaries for newly qualified accountants
fol ow a normal distribution with a standard deviation of $2,500. A random sample of 16
observations was taken. Find the probability that the sample standard deviation is more than $3,000. 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu
Question 5. (2 marks) Consumption of alcoholic beverages by young women of drinking
age is of concern in the UK and some other European countries. Annual consumption data
(in litres) are shown below for a sample of 20 European young women. 266 82 199 174 97 170 222 115 130 169 164 102 113 171 0 93 0 93 110 130
a. (1 mark) Calculate the sample mean and variance.
a. (1 mark) Assuming normality, construct a 95% confidence interval for the mean annual
consumption of alcoholic beverages by young European women.
Question 6. (2 marks) A random sample of 50 university admissions officers was asked
about expectations in application interviews. Of these sample members, 28 agreed that the
interviewer usual y expects the interviewee to have volunteer experience doing community
projects. Test that the population proportion is larger than one-half. Use  = 0.05. 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu TEST 5 Time: 60 minutes
Question 1. (2 marks) Of 100 patients with a certain disease, 10 were chosen at random to
undergo a drug treatment that increases the cure rate from 50% for those not given the
treatment to 75% for those given the drug treatment.
a. (1 mark) What is the probability that a randomly chosen patient both was cured and was given the drug treatment?
b. (1 mark) What is the probability that a patient who was cured had been given the drug treatment?
Question 2. (2 marks) The amount of time devoted to studying statistics each week by
students who achieve a grade of A in the course is a normal y distributed random variable
with a mean of 7.5 hours and a standard deviation of 2.1 hours.
a. (1 mark) What proportion of A students study for more than 10 hours per week?
b. (1 mark) What is the amount of time below which only 5% of al A students spend studying?
Question 3. (1 mark) We are given the fol owing probability distribution. X 0 1 2 3 P(x) 0,4 0,3 0,2 0,1
Suppose that Y = 3X + 2. What is the probability distribution of Y ?
Question 4. (4 marks) Dr. Susan Benner is an industrial psychologist. She is currently
studying stress among executives of Internet companies. She has developed a questionnaire
that she believes measures stress. A score above 80 indicates stress at a dangerous level. A
random sample of 15 executives revealed the fol owing stress level scores.
94, 78, 83, 90, 78, 99, 97, 90, 93, 100, 75, 84
a. (1 mark) Calculate the sample mean and variance.
b. (1 mark) Assume normality, construct a 98 percent confidence level for the population mean. 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu
c. (2 marks) Test that Internet executives have a mean stress level in the dangerous level,
according to Dr. Benner’s test? Use  = 0.05.
Question 5. (1 mark) A random sample of 100 men contained 61 in favor of a state
constitutional amendment to retard the rate of growth of property taxes. An independent
random sample of 100 women contained 54 in favor of this amendment. A confidence
interval extending from 0.04 to 0.10 was calculated for the difference between the
population proportions. Determine the confidence level of this interval. 22:15, 27/01/2026
Sample Final Exam XSTK: Statistics and Probability Analysis - Studocu