FE Mechanical Probability and Statistics

Knowledge area 2 of the FE Mechanical exam: 4–6 of the 110 questions. The subtopics NCEES lists, how they are tested and worked practice problems with solutions.

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110 questions on the real 5 hours 20 minutes clock, weighted to the NCEES specifications, with a score by knowledge area and a worked solution for every answer.

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What the specification lists

Probability distributions (e.g., normal, binomial, empirical, discrete, continuous)
Measures of central tendencies and dispersions (e.g., mean, mode, standard deviation, confidence intervals)
Expected value (weighted average) in decision making
Regression (linear, multiple), curve fitting, and goodness of fit (e.g., correlation coefficient, least squares)

From the NCEES FE Mechanical CBT exam specifications, knowledge area 2.

Line drawing of a bell curve on a whiteboard, a pair of dice and a scatter of points with a trend line

How Probability and Statistics is tested

Expect a mean, a standard deviation, a normal-distribution probability with a z value, a confidence interval or a regression line. Practice reading the unit normal table in the Reference Handbook section on engineering probability and statistics, and know when to use the sample (n − 1) rather than the population standard deviation.

On exam day the relationships you need are in the Reference Handbook section on Engineering Probability and Statistics. The handbook is the only reference allowed, so practice finding each formula in it rather than memorizing a separate sheet.

Start the free Probability and Statistics drill

FE Mechanical Probability and Statistics practice problems

Work each problem with your calculator and the handbook, then open the solution.

Problem 1 · Expected value and expected error in decision making

A buried water main has a 0.03 annual probability of failure, and a failure causes $1.2 million in damage. A monitoring system costs $15,000 per year and reduces the annual failure probability to 0.01. The expected annual net savings from installing the monitoring system, after paying its $15,000 annual cost, is most nearly:

A.$9,000
B.$21,000
C.$24,000
D.$27,000

Show the answer and solution

Answer: A: $9,000

Without monitoring the expected annual cost is 0.03($1.2M) = $36,000. With monitoring it is $15,000 + 0.01($1.2M) = $27,000. Savings = $36,000 − $27,000 = $9,000 per year. The $24,000 risk reduction must still pay for the $15,000 monitoring cost.

Expected value and expected error in decision making · Reference Handbook: Engineering Probability and Statistics

Problem 2 · Regression (linear, multiple), curve fitting, and goodness of fit

A least-squares line y = a + bx is fitted to 8 data points with Σx = 56, Σy = 148.0, Σxy = 1,156, and Σx² = 476. Enter the intercept a to two decimal places.

Fill in the blank: answer in .

Show the answer and solution

Answer: 8.5 (accepted 8.4 to 8.6)

b = (nΣxy − ΣxΣy)/(nΣx² − (Σx)²) = (9,248 − 8,288)/(3,808 − 3,136) = 960/672 = 1.4286. a = ȳ − b·x̄ = 18.50 − 1.4286(7.0) = 8.50. The line passes through the point of means (x̄, ȳ), which is how a follows from b.

Regression (linear, multiple), curve fitting, and goodness of fit · Reference Handbook: Engineering Probability and Statistics

Problem 3 · Measures of central tendencies and dispersions

Select all that apply.

Seven bridge-deck inspections took 12, 15, 15, 18, 20, 22, and 45 hours. Select all statements that are true for this data set.

A.The mean is 21 h.
B.The median is 20 h.
C.The mode is 15 h.
D.The range is 33 h.
E.Removing the 45 h value changes the mode.

Show the answer and solution

Answer: A: The mean is 21 h.; C: The mode is 15 h.; D: The range is 33 h.

Mean = 147/7 = 21 h. The ordered middle (4th) value is 18 h, so the median is 18 h, not 20 h. The mode is 15 h (it occurs twice), and the range is 45 − 12 = 33 h. Dropping 45 h leaves 15 h as the only repeated value, so the mode does not change.

Measures of central tendencies and dispersions · Reference Handbook: Engineering Probability and Statistics

Problem 4 · Estimation

Thirty-six anchors from a large lot are pull-tested, giving a sample mean of 52.4 kN. The population standard deviation is known to be 4.2 kN. With z(0.025) = 1.96, the 95% confidence interval for the lot's mean capacity is:

A.44.2 kN to 60.6 kN
B.51.0 kN to 53.8 kN
C.51.3 kN to 53.6 kN
D.52.2 kN to 52.6 kN

Show the answer and solution

Answer: B: 51.0 kN to 53.8 kN

Half-width = zσ/√n = 1.96(4.2)/√36 = 1.96(0.70) = 1.37 kN, so the interval is 52.4 ± 1.37 = 51.0 kN to 53.8 kN. Omitting √n gives the much wider 44.2 kN to 60.6 kN, a range for individual anchors rather than for the mean.

Estimation · Reference Handbook: Engineering Probability and Statistics

More FE Mechanical practice problems from every knowledge area

Sit the full exam under the real clock

Take a full timed FE Mechanical practice exam

110 questions on the real 5 hours 20 minutes clock, weighted to the NCEES specifications, with a score by knowledge area and a worked solution for every answer.

Or start the free Probability and Statistics drill

Common questions

How many Probability and Statistics questions are on the FE Mechanical exam?

The NCEES FE Mechanical specification lists 4–6 questions on Probability and Statistics, out of 110. Every examinee gets the same number per knowledge area.

What does Probability and Statistics cover on the FE Mechanical exam?

Probability distributions (e.g., normal, binomial, empirical, discrete, continuous); Measures of central tendencies and dispersions (e.g., mean, mode, standard deviation, confidence intervals); Expected value (weighted average) in decision making; Regression (linear, multiple), curve fitting, and goodness of fit (e.g., correlation coefficient, least squares).

Where are the Probability and Statistics formulas on exam day?

In the NCEES FE Reference Handbook, which appears on screen during the exam: the Engineering Probability and Statistics section. Practice with your free copy from MyNCEES so you know where each relationship sits.

Original independent practice questions written to the published NCEES exam specifications. Not affiliated with or endorsed by NCEES (National Council of Examiners for Engineering and Surveying).