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

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)

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))

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.

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

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

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.

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

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

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

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

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

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

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

Probability density functions

For any PDF P(a < x < b) = area under f(x) between a and b

Normal PDF

Area under any f(x) is 1

Normal PDFs with different µ and σ2

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

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

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

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

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

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

15. Given the following pairs of data points:

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

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

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.

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?

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

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

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

engineering probability and statistics professor j. …...other disciplines fe specifications topic:...

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