Statistical Estimation and Hypothesis Testing

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

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)

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

• 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.