Engineering Probability and Statistics Professor J. …...Other Disciplines FE Specifications Topic:...
Transcript of Engineering Probability and Statistics Professor J. …...Other Disciplines FE Specifications Topic:...
Fundamentals of Engineering Exam Review
Engineering Probability and Statistics
Professor J. B. O’Neal
Fundamentals of Engineering Exam Review
Other Disciplines FE Specifications
Topic: Engineering Probability and Statistics
6–9 FE exam problems
Exam Problem Numbers
A. Measures of central tendencies and dispersions (e.g., mean, mode, variance, standard deviation)
1-3
B. Probability distributions (e.g., discrete, continuous, normal, binomial)
4-14,16,19, 21
C. Estimation (e.g., point, confidence intervals) 18-19
D. Expected value (weighted average) in decision making 15
E. Sample distributions and sizes 20
F. Goodness of fit (e.g., correlation coefficient, least squares))
15
Fundamentals of Engineering Exam Review
We are grateful to NCEES for granting us
permission to copy short sections from the
FE Handbook to show students how to use
Handbook information in solving problems.
This information will normally appear in
these videos as white boxes.
Fundamentals of Engineering Exam Review
1. Three samples, 2,1, and 6 are taken from a large population.
The sample standard deviation is most nearly:
(A) 1.00 (B) 1.52 (C) 2.16 (D) 2.65
Fundamentals of Engineering Exam Review
2. A certain population has a mean μ and standard deviation σ. We
wish to estimate the mean of this population from a sample size n
taken from this population. We can increase the accuracy of our
estimate by increasing:
(A) μ (B) σ (C) σ2 (D) n
Fundamentals of Engineering Exam Review
3. Assume these data are samples taken from a large
population: 4, 4, 3, 5, 4, 6, 3. Find the median, mode, and
the range of these samples.
Fundamentals of Engineering Exam Review
4. An urn contains 50 balls numbered 1-50.The
probability that a ball taken from this urn is even
and greater than 30 is:
(A) 0.10 (B) 0.15 (C) 0.20 (D) 0.25
Fundamentals of Engineering Exam Review
5. A fair coin is flipped six times. What is the probability
that tails will appear at least once?
(A) 0.961 (B) 0.969 (C) 0.975 (D) 0.984
Fundamentals of Engineering Exam Review
6. The probability that no two people in a group of six
have the same birthday in a given year is most nearly:
(A) 0.92 (B) 0.94 (C) 0.96 (D) 0.98
Fundamentals of Engineering Exam Review
7. The reliabilities of the first and second stages of a rocket are
independent. If the reliability of the two stage rocket is required to be .96
and the reliability of the second stage is .98, what is the required
reliability of the first stage:
(A) 0.92 (B) 0.94 (C) 0.96 (D) 0.98
Fundamentals of Engineering Exam Review
8. Two marksmen are shooting at the same target.
The probability that the first will hit the target is 0.5
and, for the second, it is 0.7. If each fires one shot,
the probability that at least one will hit the target is
most nearly:
(A) 0.75 (B) 0.80 (C) 0.85 (D) 0.90
Fundamentals of Engineering Exam Review
9. The probability is 0.3 that a fish caught on a certain pier is a whiting. You catch five fish.
What is the probability that:
(A) the first three you catch are whiting? (B) none are whiting?
(C) you catch exactly 3 whiting?
(D) the first 3 are whiting and the others are not?
(E) 3 or more are whiting?(2 or fewer are not whiting)Use the Cumulative Binomial Probabilities P(X ≤ x) table with n=5, x=2, p=0.7
(x=2 successes in n=5 trials) See table
Fundamentals of Engineering Exam Review
Fundamentals of Engineering Exam Review
10. The probability that exactly 3 people in a group of 8
were born on a Tuesday is most nearly:
(A) .076 (B) .096 (C) .116 (D) .120
Fundamentals of Engineering Exam Review
Probability density functions
f(x)
a b
For any PDF P(a < x < b) = area under f(x) between a and b
Fundamentals of Engineering Exam Review
Normal PDF
f(x)
Area under any f(x) is 1
Fundamentals of Engineering Exam Review
Normal PDFs with different µ and σ2
Fundamentals of Engineering Exam Review
Unit Normal PDF
-3 -2 -1 0 1 2 3
Example: using the table
P(x > 0.2) = R(0.2) = 0.4207
Table exists for larger values of x
f(x)
Fundamentals of Engineering Exam Review
When f(x) is normal but not unit normal
-3 -2 -1 0 1 2 3
Unit normal fn(z)Any normal pdf, f(x)
x1 convert to z1=(x1-µ)/σ
P(x>x1) = P(z>z1) = R(z1)
Example: suppose that f(x) is normal with µ=8 and σ =4
We want to know P(x>10). x1=10, so z1=(10-8)/4) =0.5
So P(x>10) = P(z>0.5) = R(0.5) = 0.3085 from handbook unit normal table
z
Fundamentals of Engineering Exam Review
11. A certain population has a normal probability density function with mean 0 and
variance 1. The probability that a single observation taken from this population is
between -1 and +2 is most nearly:
(A) 0.62 (B) 0.72 (C) 0.82 (D) 0.92
Fundamentals of Engineering Exam Review
12. A certain population has a normal probability density function with
mean -2 and standard deviation 3. The probability that a single
observation taken from this population is greater than +3 is most nearly:
(A) .015 (B) .025 (C) .035 (D) .047
Fundamentals of Engineering Exam Review
13. The true weight of a certain object is equally likely to be any value
between 50 and 60 lbs. What is the probability that the observed weight of
the object is between 52 and 52.5 lbs.?
(A) 0.05 (B) 0.10 (C) 0.15 (D) 0.20
Fundamentals of Engineering Exam Review
14. A certain random variable x has a probability density function f(x) which
is a triangle with vertices at (5,0), (5,1/2), and (9,0).What is the probability
that the random variable is greater than 7?
(A) 1/4 (B) 1/8 (C) 1/16 (D) 1/24
Fundamentals of Engineering Exam Review
15. Given the following pairs of data points:
x y
4 2
5 2
6 3.5
0.5 -2.5
From linear least squares regression, the equation that best fits this data is:
(A) y = x + 2.9 (B) y = -x + 2.9 (C) y = x - 2.9 (D) y = -x - 2.9
Fundamentals of Engineering Exam Review
16. The random variable x has an exponential distribution
with a variance of 4. The mean value of x is most nearly:
(A)1 (B)2 (C) 3 (D)4
Fundamentals of Engineering Exam Review
17. Which of the following is a restatement of the Central Limit
Theorem:
(A)The mean value of n samples of a random variable approaches the
population mean as n increases.
(B) The number of permutations of n objects taken r at a time is n!/(n-r)!
(C) P(A+B) = P(A) + P(B) – P(A,B)
(D) As n gets large, the sum of n independent random samples taken
from the same population approaches a normal distribution.
Fundamentals of Engineering Exam Review
18. Two six sided dice are rolled. Each die contains one of the numbers 1-6.
What is the probability that the sum of the two dice is equal to 11?
Fundamentals of Engineering Exam Review
19. A random process has a normal distribution with mean 5 and standard deviation 3.
The probability that a sample from this process has a magnitude that is within 2
standard deviations of its mean is most nearly.
(A) 0.9 (B) 0.93 (C) 0.95 (D) 0.97
Fundamentals of Engineering Exam Review
20. The probability is 0.3 that a fish caught on a certain pier is a
whiting. If you catch 5 fish, the expected value of the number of whiting that you catch is most nearly:
(A) 1.5 (B) 1.7 (C) 1.9 (D) 2.0
Fundamentals of Engineering Exam Review
21. What is K for this probability density function: f(x) = K 𝑒−¼ 𝑥−3 2
(A) 3/4 (B) 1/ 2√ π (C) 3 (D) 4