A sample of five observations is 4, 7, 7, 9, 13. What is the median of the sample?
- 7 Correct answer.
- 8
- 7.5
- 9
Explanation
The median is the middle value of the ordered data. With five ordered observations, the third value is the median, which is 7.
FE Industrial & Systems practice questions
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Events A and B are independent with P(A) = 0.4 and P(B) = 0.5. What is P(A or B)?
- 0.9
- 0.2
- 0.6
- 0.7 Correct answer.
Explanation
The addition rule gives P(A or B) = P(A) + P(B) − P(A and B). For independent events, P(A and B) = 0.4 × 0.5 = 0.2, so the answer is 0.4 + 0.5 − 0.2 = 0.7.
How many different ways can a committee of 3 people be selected from a group of 8 people?
- 24
- 512
- 336
- 56 Correct answer.
Explanation
Because the order of selection does not matter, the count is the combination C(8,3) = 8!/(3!5!) = 56.
For a standard normal distribution, what is the area to the left of z = 1.00?
- 0.8413 Correct answer.
- 0.1587
- 0.3413
- 0.5000
Explanation
The standard normal table gives the cumulative area to the left of z = 1.00 as about 0.8413. By symmetry the tail area beyond z = 1.00 is about 0.1587.
If f(x) = 3x^2 − 5x + 2, what is f'(x)?
- 3x − 5
- 6x − 5 Correct answer.
- 6x^2 − 5
- 6x + 2
Explanation
The power rule gives the derivative of 3x^2 as 6x and the derivative of −5x as −5. The constant 2 differentiates to zero, leaving 6x − 5.
What is the determinant of the matrix with first row [4, 3] and second row [2, 5]?
- 14 Correct answer.
- 7
- 26
- 20
Explanation
For a 2×2 matrix the determinant is ad − bc, where a = 4, b = 3, c = 2, d = 5. This gives 4(5) − 3(2) = 20 − 6 = 14.
A 95% confidence interval for a population mean is computed as (12.4, 15.6). Which statement best describes the meaning of the 95% confidence level?
- In repeated sampling, about 95% of intervals constructed this way contain the true mean Correct answer.
- About 95% of the individual observations lie between 12.4 and 15.6
- About 95% of sample means fall between 12.4 and 15.6
- The population mean has a 95% probability of equaling 14.0
Explanation
The confidence level describes the long-run success rate of the procedure that builds the interval. It is not a probability statement about a fixed parameter, nor a statement about individual observations.
A hypothesis test returns a p-value of 0.03. At a significance level of 0.05, what is the appropriate conclusion?
- Reject the null hypothesis only if the significance level is changed to 0.01
- Accept the null hypothesis with 97% confidence
- Reject the null hypothesis; the result is statistically significant at the 5% level Correct answer.
- Fail to reject the null hypothesis because 0.03 is smaller than 0.05
Explanation
The decision rule rejects the null hypothesis when the p-value is less than or equal to the significance level. Since 0.03 < 0.05, the null hypothesis is rejected and the result is declared significant.
A process produces defective items with probability 0.2, independently. In a sample of 10 items, what is the probability of exactly 2 defectives?
- 0.200
- 0.302 Correct answer.
- 0.044
- 0.677
Explanation
The count follows a binomial distribution, so P(X = 2) = C(10,2)(0.2)^2(0.8)^8. This equals 45 × 0.04 × 0.16777 ≈ 0.302.
Calls arrive at a help desk according to a Poisson process with a mean of 3 calls per hour. What is the probability of zero calls in the next hour?
- 0.2240
- 0.0030
- 0.1494
- 0.0498 Correct answer.
Explanation
For a Poisson random variable, P(X = x) = e^(−λ) λ^x / x!. With λ = 3 and x = 0, the probability is e^(−3) ≈ 0.0498.
In a simple linear regression of y on x, the estimated slope is 2.5. How should this slope be interpreted?
- The correlation between x and y equals 2.5
- For each one-unit increase in x, predicted y increases by 2.5 units Correct answer.
- For each one-unit increase in y, x increases by 2.5 units
- About 2.5% of the variation in y is explained by x
Explanation
The slope measures the change in the fitted response per one-unit change in the predictor. A slope of 2.5 means the predicted value of y rises 2.5 units for each additional unit of x.
A regression model has a coefficient of determination R^2 = 0.81. What does this indicate?
- 81% of the observations lie exactly on the fitted line
- The correlation between the variables is 0.81%
- The model's predictions are wrong 19% of the time
- About 81% of the variability in the response is explained by the regression Correct answer.
Explanation
R-squared is the proportion of total variation in the response accounted for by the fitted model. A value of 0.81 means 81% of the response variability is explained, leaving 19% unexplained.
A discrete random variable X takes values 0, 1, 2, and 3 with probabilities 0.1, 0.3, 0.4, and 0.2 respectively. What is E(X)?
- 1.2
- 2.0
- 1.5
- 1.7 Correct answer.
Explanation
The expected value is the sum of each value times its probability: 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2). This totals 0 + 0.3 + 0.8 + 0.6 = 1.7.
For the data set 2, 4, 4, 4, 5, 5, 7, 9, what is the population standard deviation?
- 2 Correct answer.
- 2.5
- 1.6
- 4
Explanation
The mean is 5, and the squared deviations sum to 32. Dividing by n = 8 gives a population variance of 4, so the standard deviation is 2.
Which statement describes the central limit theorem?
- The sample variance decreases as the sample size decreases
- The sampling distribution of the sample mean is approximately normal when the sample size is large Correct answer.
- The sample mean equals the population mean when the sample size is large
- The population must be normally distributed for the sample mean to be normal
Explanation
The central limit theorem states that sums and averages of many independent observations tend toward a normal distribution as sample size grows. This holds regardless of the shape of the population distribution in most practical cases.
A correlation coefficient of r = −0.90 between two variables indicates which of the following?
- A strong negative linear association Correct answer.
- No linear association
- A weak negative linear association
- A strong positive linear association
Explanation
The magnitude of r measures the strength of the linear association, and its sign gives the direction. A value of −0.90 is close to −1 in magnitude, so the association is strong and negative.
A one-way ANOVA compares three treatment groups with four observations each. The between-group sum of squares is 30 and the within-group sum of squares is 45. What is the F statistic?
- 3.00 Correct answer.
- 6.00
- 0.67
- 1.50
Explanation
Between-group degrees of freedom are k − 1 = 2 and within-group degrees of freedom are N − k = 12 − 3 = 9. The mean squares are 30/2 = 15 and 45/9 = 5, so F = 15/5 = 3.00.
A confidence interval for a mean has a margin of error based on the standard error of the sample mean. By what factor must the sample size be multiplied to reduce the interval width by half, assuming the same confidence level and standard deviation?
- 4 Correct answer.
- 16
- 2
- 8
Explanation
The margin of error is proportional to 1 divided by the square root of n. Halving the width requires doubling the reciprocal of the square root, which means multiplying the sample size by 4.
What are the eigenvalues of the matrix with first row [2, 1] and second row [1, 2]?
- −1 and 3
- 2 and 2
- 0 and 4
- 1 and 3 Correct answer.
Explanation
The characteristic equation is (2 − λ)^2 − 1 = 0, which expands to λ^2 − 4λ + 3 = 0. Factoring gives (λ − 1)(λ − 3) = 0, so the eigenvalues are 1 and 3.
A pooled two-sample t-test compares means from samples of size 8 and size 10. How many degrees of freedom does the test statistic have?
- 8
- 18
- 16 Correct answer.
- 17
Explanation
The pooled t-test uses n1 + n2 − 2 degrees of freedom, because one degree of freedom is lost for each estimated group mean. Here that is 8 + 10 − 2 = 16.
If $1,000 is invested at 8% simple interest per year, what total amount will have accumulated after 3 years?
- $1,300
- $1,080
- $1,240 Correct answer.
- $1,259.71
Explanation
Simple interest accrues only on the original principal, so the interest is I = P x i x n = 1,000 x 0.08 x 3 = $240. Adding this to the principal gives $1,240.
A machine costs $50,000, has an estimated salvage value of $5,000, and a useful life of 5 years. Using straight-line depreciation, what is the annual depreciation charge?
- $9,000 Correct answer.
- $11,000
- $10,000
- $8,000
Explanation
Straight-line depreciation spreads the depreciable basis, cost minus salvage, evenly over the life: D = (B - S)/N = (50,000 - 5,000)/5 = $9,000 per year.
A project requires an initial investment of $30,000 and is expected to produce net cash inflows of $6,000 per year for 8 years. What is the simple payback period?
- 6 years
- 3 years
- 4 years
- 5 years Correct answer.
Explanation
The simple payback period is the initial investment divided by the uniform annual net cash flow: 30,000/6,000 = 5 years. It measures how long it takes to recover the original outlay, without discounting.
Which statement best describes the minimum attractive rate of return (MARR) used in engineering economic analysis?
- The rate a bank charges on short-term business loans
- The internal rate of return of the best project available
- The average return earned by the firm's past projects
- The minimum return a proposal must earn to be worth doing Correct answer.
Explanation
The MARR is the hurdle rate a proposed investment must meet or exceed to be acceptable. It is set to cover the cost of capital plus an allowance for the risk of the specific project, so it represents a minimum acceptable return rather than a historical average.
A nominal annual interest rate of 12% is compounded monthly. What is the effective annual interest rate?
- 12.75%
- 12.68% Correct answer.
- 12.00%
- 12.36%
Explanation
The effective annual rate is i_eff = (1 + r/m)^m - 1 with r = 0.12 and m = 12. This gives (1.01)^12 - 1 = 0.1268, or 12.68% per year.
How much will $5,000 accumulate to after 10 years at 6% interest compounded annually?
- $8,000
- $8,954 Correct answer.
- $5,300
- $9,434
Explanation
The single-payment compound amount factor applies: F = P(1 + i)^n = 5,000(1.06)^10. Since (1.06)^10 = 1.7908, the future amount is about $8,954.
What is the present worth of $2,000 received at the end of each year for 8 years at an interest rate of 10% per year? (Use (P/A, 10%, 8) = 5.335.)
- $12,000
- $9,600
- $16,000
- $10,670 Correct answer.
Explanation
A uniform series is converted to present worth with the factor (P/A, i, n): P = A(P/A, 10%, 8) = 2,000 x 5.335 = $10,670. The factor accounts for the time value of each of the eight payments.
A bridge requires $1,000 per year in maintenance forever, with the first payment one year from now. At an interest rate of 5% per year, what is the capitalized cost of the maintenance?
- $20,000 Correct answer.
- $10,000
- $5,000
- $15,000
Explanation
The capitalized cost of a perpetual uniform series is the annual amount divided by the interest rate: CC = A/i = 1,000/0.05 = $20,000. This is the present sum that would fund the payments indefinitely.
Two mutually exclusive alternatives both have 8-year lives and zero salvage. Project A costs $10,000 and returns $2,500 per year; Project B costs $17,000 and returns $4,000 per year. With MARR = 10% and (P/A, 10%, 8) = 5.335, which alternative should be selected by incremental analysis?
- B, since the incremental investment beats the MARR Correct answer.
- A, because it requires the smaller initial investment
- Neither, since the incremental benefit is negative
- A, because it has the higher internal rate of return
Explanation
Incremental analysis compares B minus A: an extra $7,000 of investment yields an extra $1,500 per year for 8 years. The incremental present worth is 1,500(5.335) - 7,000 = +$3,003, so the increment earns more than the MARR and B should be chosen.
Equipment costing $80,000 has a 4-year life and is depreciated by the double declining balance method with no salvage value. What is the depreciation charge in the first year?
- $20,000
- $64,000
- $16,000
- $40,000 Correct answer.
Explanation
The double declining balance rate is 2/N = 2/4 = 50% of book value each year. First-year depreciation is therefore 0.50 x $80,000 = $40,000.
A project is expected to earn a nominal return of 10% per year while general inflation is 4% per year. What is the real (inflation-free) rate of return?
- 6.00%
- 5.77% Correct answer.
- 14.40%
- 4.00%
Explanation
The Fisher relationship gives the real rate as (1 + i)/(1 + f) - 1 = 1.10/1.04 - 1 = 0.0577. The real rate must be found by division, not by subtracting the inflation rate.
A product sells for $27 per unit, has a variable cost of $15 per unit, and incurs fixed costs of $120,000 per year. What is the break-even quantity per year?
- 8,000 units
- 12,000 units
- 10,000 units Correct answer.
- 6,000 units
Explanation
Break-even occurs where total revenue equals total cost. Solving 27Q = 15Q + 120,000 gives Q = 120,000/12 = 10,000 units per year.
A public project has a present worth of benefits of $600,000 and a present worth of costs of $480,000. What is the conventional benefit-cost ratio?
- 1.20
- 1.25 Correct answer.
- 0.80
- 2.25
Explanation
The conventional benefit-cost ratio is the present worth of benefits divided by the present worth of costs: 600,000/480,000 = 1.25. A ratio greater than 1.0 indicates the project is economically acceptable.
A firm spent $50,000 last year on a completed feasibility study for a proposed plant. In deciding now whether to build the plant, how should the $50,000 be treated?
- A sunk cost that should be ignored in the decision Correct answer.
- A capital cost eligible for depreciation this year
- An incremental cost that must be recovered first
- A relevant opportunity cost that should be included
Explanation
Only future costs and revenues that differ among alternatives are relevant to a decision. Money already spent and unrecoverable is a sunk cost, so it must be excluded when comparing the alternatives available now.
A project produces $1,000 in year 1 and the annual cash flow increases by $500 each year through year 4, giving flows of $1,000, $1,500, $2,000, and $2,500. At 10% interest, what is the present worth?
- $6,000
- $4,800
- $5,359 Correct answer.
- $5,000
Explanation
Each cash flow must be discounted individually: 1,000/1.10 + 1,500/1.21 + 2,000/1.331 + 2,500/1.4641. The sum is $909 + $1,240 + $1,503 + $1,708 = $5,359.
A project requires $20,000 now and returns $5,000 per year for 6 years. What is the internal rate of return (to the nearest 0.5%)?
- 15.0%
- 11.5%
- 13.0% Correct answer.
- 10.0%
Explanation
The IRR makes the present worth zero, so 5,000(P/A, i, 6) = 20,000, or (P/A, i, 6) = 4.000. The factor equals 4.000 at about 13% (3.998), so the IRR is approximately 13%.
A project requires a $100,000 investment depreciated by straight line over 4 years to zero salvage. Annual revenue is $60,000, annual operating cost is $20,000, and the tax rate is 25%. What is the annual after-tax cash flow in years 1 through 4?
- $30,000
- $26,250
- $36,250 Correct answer.
- $40,000
Explanation
Annual depreciation is $25,000, so taxable income is 60,000 - 20,000 - 25,000 = $15,000 and tax is $3,750. After-tax cash flow equals net income plus depreciation: 11,250 + 25,000 = $36,250, or equivalently 40,000(0.75) + 25,000(0.25).
A defender has a current market value of $8,000, annual operating cost of $6,000, a remaining life of 3 years, and zero salvage. A challenger costs $25,000, has an annual operating cost of $3,000, a 10-year life, and zero salvage. With MARR = 10%, (A/P, 10%, 3) = 0.4021 and (A/P, 10%, 10) = 0.1627, what action is justified?
- Keep the defender one more year and then replace it
- Be indifferent, because both alternatives have zero salvage
- Replace now, because the challenger's annual worth is higher Correct answer.
- Keep the defender, because it needs no new capital outlay at all
Explanation
Annual worth of the defender is -8,000(0.4021) - 6,000 = -$9,217, and of the challenger is -25,000(0.1627) - 3,000 = -$7,069. Because the challenger has the higher (less negative) annual worth, it should replace the defender now.
A project has cash flows of -$5,000 at time 0, +$25,000 at time 1, and -$20,000 at time 2. How many internal rates of return does this cash flow series have?
- None, because the cash flows change sign
- Two, at 0% and 300% Correct answer.
- One, at 100%
- One, at 300%
Explanation
The series has two sign changes, so by Descartes' rule up to two positive roots may exist. Solving -5,000(1+i)^2 + 25,000(1+i) - 20,000 = 0 gives roots of 1.0 and 4.0 for (1+i), so the IRRs are 0% and 300%.
A project requires $50,000 now and returns $20,000 per year for 4 years. Both the reinvestment rate and the borrowing rate are 10%, and (F/A, 10%, 4) = 4.641. What is the external rate of return (ERR)?
- 13.0%
- 20.0%
- 16.7% Correct answer.
- 18.5%
Explanation
The ERR compounds all inflows forward to year 4 at the reinvestment rate: 20,000(4.641) = $92,820. Setting 50,000(1 + ERR)^4 = 92,820 gives (1 + ERR)^4 = 1.8564, so ERR = 16.7%.
Under the NCEES Code of Ethics, which obligation takes precedence over the others?
- Safeguard confidential business information of the employer
- Enhance the reputation of the engineering profession
- Protect public health, safety, and welfare Correct answer.
- Maximize the financial return of the client or employer
Explanation
The first and paramount canon of engineering ethics is to hold the safety, health, and welfare of the public above all other interests. Client profitability, employer confidentiality, and professional reputation are important but subordinate to public protection.
An engineer is asked to design a foundation system but has experience only in HVAC design. What is the most appropriate action?
- Proceed without informing the client of limitations
- Sign the drawings and let a colleague check them later
- Obtain qualified help or decline the work Correct answer.
- Accept the work and rely on published design tables
Explanation
Engineers must perform services only in areas where they are competent. If the engineer lacks the needed expertise, the proper course is to obtain qualified assistance or decline the assignment. Concealing a competence gap from the client is deceptive.
Under professional licensing rules, an engineer may seal engineering drawings only when:
- The engineer prepared or directly supervised the work Correct answer.
- Another licensed engineer has reviewed the drawings
- The firm holds a valid contract with the owner
- The client requests a seal for permitting
Explanation
The seal certifies that the engineer was in responsible charge of the work and accepts professional responsibility for it. Reviewing someone else's drawings, holding the contract, or responding to a client request does not by itself justify sealing.
An engineer may disclose a client's confidential technical information without consent when:
- The engineer accepts a position with a competitor
- The records are more than ten years old
- A colleague requests benchmarking data from the project
- Law requires it or public safety demands it Correct answer.
Explanation
Confidential client information must be protected unless disclosure is legally required or necessary to protect public health and safety. Neither the passage of time nor a change of employment removes that obligation.
Accepting compensation from both the owner and the contractor on the same project is permitted only when:
- All parties disclose and agree in writing Correct answer.
- The contractor is a licensed engineering firm
- The owner gives verbal approval for the arrangement
- The engineer's fee is below the market rate
Explanation
Rules of professional conduct prohibit undisclosed dual compensation on the same project. The conflict may be resolved only if the circumstances are fully disclosed and all interested parties agree in writing.
Which statement best describes ethical advertising by an engineering firm?
- Designed to exaggerate the firm's capabilities
- Limited to claims that cannot be verified
- Accurate, truthful, and not misleading Correct answer.
- Intended to disparage competing firms
Explanation
Professional advertising must be objective and truthful and must not create misleading impressions. Exaggeration, unverifiable claims, and disparagement of competitors are deceptive practices.
A PE's supervisor directs her to approve a pressure vessel design that she believes violates code. After raising concerns internally, the supervisor insists. What should she do?
- Approve it and document her objection in a private file
- Resign quietly and say nothing
- Approve it because the supervisor is responsible
- Refuse to approve and notify the appropriate authority Correct answer.
Explanation
When an engineer's professional judgment is overruled in a way that endangers public safety, the engineer must refuse to certify the work and notify the appropriate authority. Private documentation or silent resignation does not protect the public.
A vendor whose products the engineer specifies offers an all-expenses-paid vacation. What is the appropriate response?
- Accept because small gifts do not influence decisions
- Accept and report it to the client later
- Decline it; it may improperly influence selection Correct answer.
- Accept if the vendor is not selected yet
Explanation
Engineers may not solicit or accept valuable consideration from parties dealing with their clients in connection with work for which the engineer is responsible. A vacation from a prospective vendor creates an improper inducement and should be declined.
An engineer is hired as an expert witness in a dispute. The engineer should:
- Advocate strongly for the party that retained her
- Avoid disclosing compensation for testimony
- Offer opinions beyond expertise if the fee is high
- Testify objectively within areas of competence Correct answer.
Explanation
Expert testimony must be objective and confined to matters within the engineer's technical competence. Compensation may be disclosed and does not justify advocacy or opinions outside the engineer's expertise.
A junior engineer prepares calculations, and the PE briefly reviews them before sealing the drawings. This practice is:
- Unethical only if an error is later found
- Acceptable if the junior has an engineering degree
- Acceptable because the PE reviewed the work
- Unethical unless the PE had responsible charge Correct answer.
Explanation
The seal certifies responsible charge, meaning the engineer personally prepared or directly supervised and controlled the work. A brief review alone does not establish responsible charge and does not justify sealing.