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Linearized Form

Environmental Engineering · FE Reference Handbook section

Environmental Engineering
1 formulas
1 exam-style examples
~20 min
All Environmental Engineering lectures

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1
Least-squares slope for a settlement trend

Settlement readings (t, s) are (1, 4), (2, 7), (3, 9) and (4, 12) mm. Fit s = a + bt by least squares.

Given

  • n=4n = 4

  • Σt = 10, Σs = 32, Σts = 95, Σt² = 30

Find

a and b

Start with the thinking

  • Compute the sums before touching the formulas.
  • Slope first; the intercept follows from the means.

Step-by-step solution

  1. Slope — b = (nΣts − ΣtΣs)/(nΣt² − (Σt)²)

  2. Numerator

    4(95)−(10)(32)=380−320=60.04(95) - (10)(32) = 380 - 320 = 60.0

  3. Denominator

    4(30)−100=20.04(30) - 100 = 20.0

  4. Slope

    b=60.0/20.0=3.00mmperperiodb = 60.0/20.0 = 3.00 mm per period

  5. Intercept

    a=sˉ−btˉ=8.00−3.00(2.50)=0.500mma = s̄ - b t̄ = 8.00 - 3.00(2.50) = 0.500 mm
Answer:
s=0.500+3.00t(mm)s = 0.500 + 3.00t (mm)

Why the other options are there

  • b = 0.333 (ratio inverted)
  • a = 8.00 (mean reported as intercept)

Reference: FE Reference Handbook — Probability and Statistics — Linear regression

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