---
title: "Regression of permeability on compressive strength"
subject: "Probability and Statistics"
chapter: 5
key: "semester_5/probability_and_statistics:concrete-permeability-regression-sums"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/concrete-permeability-regression-sums
site: BCT Engineering
---

# Regression of permeability on compressive strength

Probability and Statistics · Chapter 5: Correlation and regression

- Asked in: 2069 Chaitra, 6 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/concrete-permeability-regression-sums
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5

## Question

An article in Concrete Research presented data on compressive strength X and intrinsic permeability Y of various concrete mixes and cures. Summary quantities are n = 14, Σy = 572, Σy² = 23,530, Σx = 43, Σx² = 157.42 and Σxy = 1697.80. Assume that the two variables are related according to the simple linear regression model.
i) Calculate the least squares estimates of the slope and intercept
ii) Use the equation of the fitted line to predict what permeability would be observed when the compressive strength is x = 4.3.

## Short answer

i) β̂₁ = −2.330, β̂₀ = 48.013 so ŷ = 48.013 − 2.330x. ii) At x = 4.3, predicted permeability ≈ 38.0.

## Answer

Model: Y = β₀ + β₁X + ε, with X = compressive strength and Y = permeability. Given n = 14, Σx = 43, Σy = 572, Σx² = 157.42, Σy² = 23530, Σxy = 1697.80.

#### i) Least squares estimates

Means: x̄ = 43/14 = 3.0714, ȳ = 572/14 = 40.8571

- Sxx = Σx² − (Σx)²/n = 157.42 − 43²/14 = 157.42 − 132.0714 = 25.3486
- Sxy = Σxy − (Σx)(Σy)/n = 1697.80 − (43)(572)/14 = 1697.80 − 1756.8571 = -59.0571

Slope: β̂₁ = Sxy/Sxx = -59.0571/25.3486 = **-2.3298**

Intercept: β̂₀ = ȳ − β̂₁x̄ = 40.8571 − (-2.3298)(3.0714) = **48.0130**

Fitted line: **ŷ = 48.013 − 2.330 x**

#### ii) Prediction at x = 4.3

ŷ = 48.0130 + (-2.3298)(4.3) = 48.0130 − 10.0181 = **37.99**

**Answer: β̂₁ = -2.330, β̂₀ = 48.013; predicted permeability at x = 4.3 is about 38.0.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is Sxx computed from the sums?
   A. Σx² − (Σx)²/n = 25.35
   B. Σx² − Σx
   C. (Σx)² − Σx²
   D. Σx²/n
   Answer: A. Sxx = 157.42 − 43²/14 = 25.3486. (Key point: Sxx)

2. Why is the slope negative?
   A. Sxy = -59.06 is negative
   B. Sxx is negative
   C. x̄ is negative
   D. n is large
   Answer: A. Sxy = 1697.80 − 1756.86 = -59.06, so β̂1 = -2.33. (Key point: Slope sign)

3. What is β̂0?
   A. -2.33
   B. 48.01
   C. 40.86
   D. 3.07
   Answer: B. β̂0 = ȳ − β̂1x̄ = 40.857 + 2.3298(3.0714) = 48.013. (Key point: Intercept)

4. Predicted permeability at x = 4.3 is about:
   A. 58.0
   B. 48.0
   C. 10.0
   D. 38.0
   Answer: D. ŷ = 48.013 − 2.330(4.3) ≈ 38.0. (Key point: Prediction)

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