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Statistical Estimation and Hypothesis Testing

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FE Fundamentals of Engineering Exam

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1 Introduction to the FE Exam

About the FE Exam FE Exam Study Strategies Navigating the FE Reference Handbook

2 Mathematics (8–12 Questions)

Analytic Geometry and Trigonometry Single-Variable Calculus Differential Equations Linear Algebra and Matrix Operations Numerical Methods

3 Probability and Statistics (6–9 Questions)

Probability Fundamentals Probability Distributions Statistical Estimation and Hypothesis Testing

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Chemical Reactions and Stoichiometry Acids, Bases, and pH Oxidation-Reduction and Corrosion

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9 Strength of Materials (9–14 Questions)

Stress and Strain Fundamentals Beam Analysis: Shear and Moment Diagrams Bending Stress and Torsion Stress Transformation and Material Failure

10 Materials Science (6–9 Questions)

Material Properties and Selection Phase Diagrams and Alloy Systems

11 Fluid Mechanics (12–18 Questions)

Fluid Properties and Statics Fluid Dynamics and Bernoulli's Equation Pumps, Turbines, and Flow Measurement

12 Thermodynamics and Heat Transfer (9–14 Questions)

Laws of Thermodynamics and Properties Thermodynamic Cycles and Applications Heat Transfer: Conduction, Convection, and Radiation

13 Basic Electrical Engineering (6–9 Questions)

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16 Computational Tools and Computer Science

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Statistical Estimation and Hypothesis Testing

Key Takeaways Explain This Topic AI

Sample variance uses (n−1) in the denominator — Bessel's correction — to give an unbiased estimate of σ².

A confidence interval for the mean is x̄ ± z(α/2)·σ/√n when σ is known, or x̄ ± t(α/2,n−1)·s/√n when σ is unknown.

Common two-sided z critical values: 1.645 (90%), 1.960 (95%), 2.576 (99%).

Use the t-distribution instead of z when σ is unknown and the sample is small (n < 30).

Type I error (α) is rejecting a true H₀; Type II error (β) is failing to reject a false H₀; power = 1 − β.

R² is the fraction of variance in y explained by the model; the correlation r = ±√(R²) carries the sign of the slope.

Last updated: September 2026

Inferential statistics draws conclusions about a population from a sample. The FE tests the mechanics : compute a sample statistic, build an interval, or run a test against a critical value. Tables of z and t critical values are in the Handbook.

Descriptive Statistics

  • Measure of center Formula Note
  • Mean x̄ Σxᵢ/n sensitive to outliers
  • Median middle of sorted data robust to outliers
  • Mode most frequent value for categorical data
  • Measure of spread Formula
  • Range max − min
  • Sample variance s² Σ(xᵢ − x̄)²/(n−1)
  • Sample std. deviation s √(s²)
  • Coefficient of variation (s/x̄)×100%

Bessel's correction: dividing the sample variance by (n−1) rather than n removes bias in estimating the population σ². Using n instead is the single most common FE statistics error.

Worked example — mean and SD. For data {2, 4, 4, 6, 9}: x̄ = 25/5 = 5. Deviations squared: (−3)², (−1)², (−1)², (1)², (4)² = 9+1+1+1+16 = 28. Sample variance s² = 28/(5−1) = 7, so s = √7 ≈ 2.65. Had you divided by n = 5 you would get 5.6, the (biased) population variance — a different and usually wrong answer on the FE.

Point Estimation

A point estimate is one number standing in for a population parameter: sample mean x̄ estimates μ, sample variance s² estimates σ², and sample proportion p̂ = x/n estimates p.

Confidence Intervals

A confidence interval (CI) gives a plausible range for a parameter at a stated confidence level.

  • σ known: x̄ ± z(α/2)·(σ/√n)
  • σ unknown (use t): x̄ ± t(α/2, n−1)·(s/√n)
  • Confidence z(α/2)
  • 90% 1.645
  • 95% 1.960
  • 99% 2.576

Use z when the population σ is known (or n is large); use the t-distribution (which has fatter tails) when σ is unknown and n is small (n < 30). The term σ/√n is the standard error of the mean — it shrinks as √n, so quadrupling the sample size only halves the interval width.

Worked example — CI with known σ. A sample of n = 36 measurements has x̄ = 82 and population σ = 12. The 95% CI is 82 ± 1.960·(12/√36) = 82 ± 1.960·(12/6) = 82 ± 1.960·2 = 82 ± 3.92 , i.e., (78.08, 85.92). The √n = 6 in the denominator is what makes the margin small despite σ = 12.

Worked example — CI with unknown σ (t). 25 concrete cylinders give x̄ = 4,500 psi and s = 300 psi. With df = 24, t₀.₀₂₅ ≈ 2.064, so the 95% CI is 4,500 ± 2.064·(300/√25) = 4,500 ± 2.064·60 = 4,500 ± 123.8 = (4,376, 4,624) psi. Using z = 1.960 here would understate the margin — small samples with unknown σ require the wider t value.

Hypothesis Testing

Procedure: (1) state H₀ (null) and H₁ (alternative); (2) pick a significance level α, commonly 0.05; (3) compute a test statistic; (4) compare it to the critical value (or compute a p-value); (5) reject H₀ if |statistic| exceeds the critical value, otherwise fail to reject.

Test statistic for a mean: z = (x̄ − μ₀)/(σ/√n) when σ is known, or t = (x̄ − μ₀)/(s/√n) when it is not.

Two Kinds of Error

  • H₀ true H₀ false
  • Reject H₀ Type I error (α) correct (power = 1 − β)
  • Fail to reject H₀ correct Type II error (β)

Type I (α): false positive — rejecting a true null. α is the significance level you choose.

Type II (β): false negative — failing to reject a false null.

Power (1 − β): probability of correctly detecting a real effect; raising n increases power.

Lowering α to guard against false positives raises β unless you also collect more data — a trade-off the FE likes to probe conceptually.

Regression and Correlation

The least-squares line ŷ = b₀ + b₁x has slope b₁ = [nΣxᵢyᵢ − ΣxᵢΣyᵢ]/[nΣxᵢ² − (Σxᵢ)²] and intercept b₀ = ȳ − b₁x̄.

Coefficient of determination R² (0 to 1): the fraction of variance in y explained by the model. R² = 0.92 means 92% of y's variation is explained; 8% is unexplained.

Correlation coefficient r (−1 to +1): strength and direction of the linear relationship. r = +1 is perfect positive, −1 perfect negative, 0 none. They are linked by r = ±√(R²), with the sign taken from the slope. A high R² does not by itself imply causation — a classic FE conceptual trap.

Fit a line to the points (1, 2), (2, 3), (3, 5). Here n = 3, Σx = 6, Σy = 10, Σxy = (1·2)+(2·3)+(3·5) = 2+6+15 = 23, Σx² = 1+4+9 = 14. The slope is b₁ = (3 × 23 − 6 × 10)/(3 × 14 − 6²) = 9/6 = 1.5. With x̄ = 6/3 = 2 and ȳ = 10/3, the intercept is b₀ = 10/3 − 1.5 × 2 = 1/3 ≈ 0.333. Thus the fitted line is ŷ = 1/3 + 1.5x. Because the slope is positive, the correlation r is positive, and r = +√(R²). This least-squares result is the same machinery used in calibration curves for instruments — fitting a sensor's output to a known input — which is exactly how the FE frames many regression questions in the Instrumentation portion of the exam.

Reading p-values. When a problem reports a p-value instead of a critical value, the rule is simple: reject H₀ if p < α. A p-value of 0.03 against α = 0.05 means reject; the same 0.03 against a stricter α = 0.01 means fail to reject. The p-value is the probability of observing data at least as extreme as yours if H₀ were true — not the probability that H₀ is true, a distinction the FE tests directly.