--- title: "Fit straight line: humidity and moisture" subject: "Probability and Statistics" chapter: 5 key: "semester_5/probability_and_statistics:humidity-moisture-straight-line-fit" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/humidity-moisture-straight-line-fit site: BCT Engineering --- # Fit straight line: humidity and moisture Probability and Statistics · Chapter 5: Correlation and regression - Asked in: 2073 Chaitra, 5 marks - Video: [Lec-4: Linear Regression📈 with Real life examples & Calculations | Easiest Explanation](https://www.youtube.com/watch?v=zUQr6HAAKp4) — Gate Smashers - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/humidity-moisture-straight-line-fit - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5 ## Question Raw material used in the production of a synthetic fibre is stored in a place which has no humidity control. Measurements of the relative humidity in the storage place and moisture content of sample of the raw material (both in percentage) on 12 days yielded the following results: | Humidity, X | 42 | 35 | 50 | 43 | 48 | 62 | 31 | 36 | 44 | 39 | 55 | 48 | |---|---|---|---|---|---|---|---|---|---|---|---|---| | Moisture content, Y | 12 | 8 | 14 | 9 | 11 | 16 | 7 | 9 | 12 | 10 | 13 | 11 | Verify that it is reasonable to fit a straight line. Fit the straight by the method of least squares. ## Short answer A straight line is reasonable because r = 0.9144, and the significance test t = 7.144 exceeds the critical value 2.228. The least squares line is Y = -0.9495 + 0.2690X, so each 1% humidity increase raises moisture content by about 0.269%. ## Answer Let X = relative humidity (%), Y = moisture content (%), n = 12. #### Is a straight line reasonable? Plotting the points (X, Y) gives a scatter that rises from lower-left to upper-right and lies close to a straight band, so a linear model is suitable. This is confirmed numerically by the correlation coefficient: r = Sxy/√(Sxx·Syy) = **0.9144** (r² = 0.836) Test: H₀: ρ = 0 against H₁: ρ ≠ 0, α = 0.05. t = r√(n−2)/√(1−r²) = 7.144, critical value t(0.025, 10) = 2.228. Since 7.144 > 2.228, H₀ is rejected: the linear relationship is significant. #### Least squares line Y = a + bX Sums (n = 12): ```text x y xy x² y² 42 12 504 1764 144 35 8 280 1225 64 50 14 700 2500 196 43 9 387 1849 81 48 11 528 2304 121 62 16 992 3844 256 31 7 217 961 49 36 9 324 1296 81 44 12 528 1936 144 39 10 390 1521 100 55 13 715 3025 169 48 11 528 2304 121 --------------------------------------------- 533 132 6093 24529 1526 ``` Σx = 533, Σy = 132, Σxy = 6093, Σx² = 24529, Σy² = 1526 Means: x̄ = 44.4167, ȳ = 11.0000 Sxx = Σx² − (Σx)²/n = 854.9167 Sxy = Σxy − (Σx)(Σy)/n = 230.0000 Syy = Σy² − (Σy)²/n = 74.0000 Slope b = Sxy/Sxx = 230.0000/854.9167 = 0.2690 Intercept a = ȳ − bx̄ = 11.0000 − (0.2690)(44.4167) = -0.9495 **Fitted line: Ŷ = -0.9495 + 0.2690 X** Interpretation: each 1% rise in humidity raises moisture content by about 0.269%. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is it verified that a straight line is reasonable? A. Check that x̄ = ȳ B. Check the scatter lies in a rising straight band and r is large (0.914) and significant C. Check that n is 12 D. Check that Σy is large Answer: B. A scatter plot close to a straight band plus a high, significant r supports a linear model. (Key point: Checking linearity) 2. The test of r with n = 12 uses critical value: A. t(0.025, 12) = 2.179 B. t(0.025, 11) = 2.201 C. z = 1.96 D. t(0.025, 10) = 2.228 Answer: D. df = n − 2 = 10; since 7.144 > 2.228, reject H0: ρ = 0. (Key point: Significance test of r) 3. Slope b = Sxy/Sxx = 230/854.92 is: A. 3.717 B. 0.914 C. 0.269 D. -0.95 Answer: C. b = 230/854.9167 = 0.2690. (Key point: Slope) 4. Which is right for Ŷ = −0.9495 + 0.269X? A. Each 1% rise in humidity raises moisture content by about 0.269%; a = ȳ − b·x̄ B. Each 1% rise lowers moisture by 0.269% C. a = x̄ − b·ȳ D. Moisture is 0.269 at zero humidity Answer: A. The slope is the change in Y per unit X, and the intercept is ȳ − b·x̄ = 11 − 0.269(44.4167). 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