R University

Statistics Exam 2021-05-09 Exam ID 2025

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1. (a) (b) (c) (d) X (e)

2. (a) (b) (c) (d) (e) X

3. (a) X (b) (c) (d) (e)

4. (a) (b) (c) X (d) (e)

5. (a) (b) (c) (d) X (e)

6. (a) (b) (c) (d) X (e)

7. (a) (b) (c) (d) X (e)

8. (a) (b) (c) X (d) (e)

9. (a) (b) (c) (d) (e) X

10. (a) (b) (c) (d) X (e)

11. (a) (b) X (c) (d) (e)

12. (a) X (b) (c) (d) (e)

Statistics Exam: 2025 2

13. (a) (b) (c) (d) X (e)

14. (a) (b) (c) (d) (e) X

15. (a) (b) (c) (d) (e) X

16. (a) (b) (c) X (d) (e)

17. (a) (b) (c) (d) (e) X

18. (a) X (b) (c) (d) (e)

Statistics Exam: 2025 3

1. Problem
In a course of Statistics, students are allowed to take an exam three times. Suppose that
students have a chance of 28% to pass the exam on the first try. If they fail this time, they
have a probability of 36% passing the exam on the second try. In case of failing at these
two tries, they still have a probability of 58% to pass on the third one. Find the probability
that a randomly chosen student needs exactly three times to pass this course.

(a) 0.1441.
(b) 0.4383.
(c) The other anwsers are wrong.
(d) 0.2673.
(e) 0.6003.

2. Problem
For ordinary exams, universities in German use a 5-point scale to evaluate their student.
The below table
summarizes the Calculus scores (X) and the Lin- X Y
ear Algebra scores (Y) of students in a certain 1 2 3 4 5
German university (rounded to be integers). Stu- 1 0.0662 0.0850 0.0222 0.0968 0.0862
dents will fail at a course if their score is at least 2 0.0775 0.0231 0.0559 0.0233 0.0949
4. Find the probability that a student in this uni- 3 0.0747 0.0074 0.0543 0.0195 0.0168
versity will fail at Linear Algebra course if he/she 4 0.0319 0.0850 0.0179 0.0051 0.0078
has passed the Calculus course. 5 0.0410 0.0053 0.0006 0.0006 0.0010

(a) 0.5592.
(b) 0.6625.
(c) 0.4792.
(d) The other answers are wrong.
(e) 0.4199..

3. Problem
A university has investigated that 95% of their students graduate from their bachelor’s pro-
gram, but only 12% of them, who have earned the bachelor’s degree, will apply for the
master’s program. Suppose that the decision of applications is independent between stu-
dents. Choosing 12 students in this university randomly, find the probability that at least 3
of them will apply for the master’s program.

(a) 0.149.
(b) 0.359.
(c) 0.263.
(d) 0.4413.
(e) The other answers are wrong.

4. Problem

Here are the percentages of cotton in material used to manufacture men’s shirts: 35.6,
32.1, 39.5, 39.2, 30.6, 39.1, 32.8, 33.7, 38.8, 31.4. Calculate the median of these data.

(a) The other answers are wrong.

(b) 34.75.
Statistics Exam: 2025 4

(c) 34.65.
(d) 35.05.
(e) 35.15.

5. Problem

Of the registered voters in a certain community, 59.2 percent are women and 40.8 percent
are men. In the last local election, 67.9 percent of the registered female voters and 62.2
percent of the registered male voters actually voted. If a registered voter from this commu-
nity is randomly chosen, what is the probability that this person is a woman who voted in
the last election?

(a) 0.7192.
(b) 0.6653.
(c) The other answers are wrong.
(d) 0.613.
(e) 0.692.

6. Problem

Suppose that the self-study time (hour/week) of a certain student is a random variable
function f (x) where its graph is given in the figure (f (x) =
with the probability density 0, ∀x ∈/ (0, 15)). What is the probability that this student has a
self-study time between 10 and 15?

(a) 0.4794.
(b) 0.2174.
(c) 0.3579.
(d) 0.3333.
(e) The other anwsers are wrong.

7. Problem

The lifespan of a certain battery type is a random variable with a mean of 11 months and
a standard deviation of 4.6 months. Suppose that 380 batteries of this type were randomly
selected. Estimate the probability that the average lifespan is more than 11.5 months.

(a) 0.3073.
(b) 0.3119.
(c) 0.1175.
(d) 0.0171.
(e) The other answers are wrong.

8. Problem
In a certain town, there is a unique bus line to the airport. The buses depart from the station
every 12 minutes and the first depart is at 7 am every day. Suppose that a traveler arrives at
the station between 7 am and 7:54 am with the arrival time following a uniform distribution.
What is the probability that he/she has missed at least 4 departs?
Statistics Exam: 2025 5

(a) 0.4075.
(b) 0.1301.
(c) 0.3333.
(d) 0.2176.
(e) The other answers are wrong.

9. Problem
The life in hours of a 75-watt light bulb is known to be normally distributed with σ = 20
hours. A random sample of 10 bulbs has a mean life of 1020 hours. Find a 98% two-sided
confidence interval on the mean life.

(a) [978, 1008].

(b) [1025, 1055].
(c) [970, 1000].
(d) [1045, 1075].
(e) [1005, 1035].

10. Problem
Suppose that a bakery will get a profit of $15 for each sold cake. If the demand in a certain
day is given as below.
Find the average and the standard de-
Number of cakes 0.0000 1.0000 2.0000 3.0000 4.0000
viation profit of this bakery in the given
Probability 0.0273 0.1581 0.1596 0.3303 0.3247
day.

(a) µ = $41.505 and σ = $16.6381.

(b) µ = $38.2871 and σ = $19.5659.
(c) The other answers are wrong.
(d) µ = $41.505 and σ = $17.1559.
(e) µ = $45.6573 and σ = $17.1559.

11. Problem
Suppose that the number of a certain item purchased in a fashion website folllows a Poisson
distribution with a mean of 4 items per hour. If the prior information shows that there are at
least 2 items sold between 7pm to 9pm, find the probability that there are exactly 10 items
sold in this period.

(a) 0.2852.
(b) 0.0996.
(c) The other answers are wrong.
(d) 0.447.
(e) 0.0721.

12. Problem

Two random variables X ∼ N(9, 0.22 ) and Y ∼ N(8, 0.272 ). Find the expected value and
variance of Z = 3X − 4Y .

(a) E(Z ) = −5; Var (Z ) = 1.5264.

(b) E(Z ) = −5; Var (Z ) = 0.8064.
Statistics Exam: 2025 6

(c) The other answers are wrong.

(d) E(Z ) = −5; Var (Z ) = 0.4116.
(e) E(Z ) = −47; Var (Z ) = 1.5264.

13. Problem
A college professor never finishes his lecture before the end of the hour and always finishes
his lectures within 5 min after the hour. Let X be the time that elapses( between the end of
3x 2 /53 , if 0 < x < 5
the hour and the end of the lecture and suppose the pdf of X is f (x) = .
0, otherwise
Find the median of X .

(a) The other answers are wrong.

(b) 3.8141.
(c) 3.2882.
(d) 3.9685.
(e) 4.574.

14. Problem
In a certain assembly plant, five machines, B1, B2, B3, B4, and B5, make 18.6%, 69.6%,
8.4%, 0.4%, and 3%, respectively, of the products. It is known from past experience that
4.4%, 10.3%, 7%, 13.6%, and 2.9% of the products made by each machine, respectively,
are defective. A finished product is randomly selected and it turns out to be defective. Which
machine most likely procduces this item?

(a) B5.
(b) B4.
(c) B3.
(d) B1.
(e) B2.

15. Problem
The following circuit operates if and only if there is a path of functional devices from left to
right. The
probabilities that each device functions are as shown where p =
0.82. Assume that the probability that a device is functional does
not depend on whether or not other devices are functional. What is
the probability that the circuit operates?

(a) 0.7253.
(b) 0.9803.
(c) The other answers are wrong.
(d) 0.8553.
(e) 0.9942.

16. Problem
The age when smokers first start from previous studies is normally distributed with a pop-
ulation standard deviation of 1.5 years old. A sample of 25 smokers found that their mean
starting age was 14 years old. Find a 98% lower (one-sided) confidence bound of the mean.

(a) 13.598.
Statistics Exam: 2025 7

(b) 13.577.
(c) 13.385.
(d) 13.556.
(e) 13.616.

17. Problem
The following table shows the joint distribution for an exam where students have to choose
one of two questions.
The passing score is 3 points or more. 5 4 3 2 1
What is the probability that a randomly Question 1 0.0443 0.0713 0.0340 0.1032 0.1369
selected student will pass the exam? Question 2 0.1045 0.0978 0.0015 0.1322 0.2743

(a) 0.5012.
(b) 0.1107.
(c) The other answers are wrong.
(d) 0.6747.
(e) 0.1827.

18. Problem
A civil engineer is analyzing the compressive strength of concrete, which is normally dis-
tributed with σ 2 = 1435 psi2 . It is desired to estimate the compressive strength with an error
that is less than 10 psi at 92% confidence. What is the smallest sample size required?

(a) 29.
(b) 20.
(c) 35.
(d) 33.
(e) 37.