--- title: "Correlation: tree circumference and height" subject: "Probability and Statistics" chapter: 5 key: "semester_5/probability_and_statistics:trees-circumference-height-correlation" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/trees-circumference-height-correlation site: BCT Engineering --- # Correlation: tree circumference and height Probability and Statistics · Chapter 5: Correlation and regression - Asked in: 2076 Chaitra, 5 marks - Video: [Correlation Coefficient](https://www.youtube.com/watch?v=11c9cs6WpJU) — The Organic Chemistry Tutor - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/trees-circumference-height-correlation - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5 ## Question Listed below are circumference (in feet) and height (in feet) of trees in Marshall, Minnesota (base on data from "Tree Measurement" by Stenley Rice, American Biology Teacher Vol 61, No 9) | X (circ) | 1.8 | 1.9 | 1.8 | 2.4 | 5.1 | 3.1 | 5.5 | 5.1 | 8.3 | |---|---|---|---|---|---|---|---|---|---| | Y (ht) | 21.0 | 33.5 | 24.6 | 40.7 | 73.2 | 24.4 | 40.4 | 45.3 | 53.5 | a) Is there a correlation exist? b) Explain this correlation. ## Short answer Yes, a significant moderate positive correlation exists (r = 0.68, t = 2.455, p < 0.05). Taller trees tend to have larger circumference; about 46.3% of height variation is explained by circumference. ## Answer Let X = circumference and Y = height, n = 9. #### a) Does a correlation exist? Karl Pearson's r = Sxy / √(Sxx·Syy) Sums (n = 9): ```text x y xy x² y² 1.8 21 37.8 3.24 441 1.9 33.5 63.65 3.61 1122.25 1.8 24.6 44.28 3.24 605.16 2.4 40.7 97.68 5.76 1656.49 5.1 73.2 373.32 26.01 5358.24 3.1 24.4 75.64 9.61 595.36 5.5 40.4 222.2 30.25 1632.16 5.1 45.3 231.03 26.01 2052.09 8.3 53.5 444.05 68.89 2862.25 --------------------------------------------- 35 356.6 1589.65 176.62 16325 ``` Σx = 35, Σy = 356.6, Σxy = 1589.65, Σx² = 176.62, Σy² = 16325 Means: x̄ = 3.8889, ȳ = 39.6222 Sxx = Σx² − (Σx)²/n = 40.5089 Sxy = Σxy − (Σx)(Σy)/n = 202.8722 Syy = Σy² − (Σy)²/n = 2195.7156 Slope b = Sxy/Sxx = 202.8722/40.5089 = 5.0081 Intercept a = ȳ − bx̄ = 39.6222 − (5.0081)(3.8889) = 20.1463 r = 202.8722 / √(40.5089 × 2195.7156) = **0.6802** **Test of significance of r** - H₀: ρ = 0 (no linear correlation in the population) - H₁: ρ ≠ 0 (two-tailed) - Level of significance α = 0.05 - Test statistic: t = r√(n − 2)/√(1 − r²) = 0.6802×√7/√(1 − 0.4627) = **2.455** - Critical value: t(0.025, df = 7) = ±2.365 - Decision: |t| = 2.455 > 2.365, so reject H₀ (p ≈ 0.044). Conclusion: at the 5% level there is significant linear correlation between tree circumference and height. #### b) Explanation r = 0.680 is positive and moderately strong. Taller trees tend to have larger circumference: as circumference increases, height generally increases. The coefficient of determination r² = 0.463, so about 46.3% of the variation in height is explained by the linear relationship with circumference; the remaining 53.7% is due to other factors (age, species, soil, etc.). Correlation does not by itself prove that a larger circumference causes greater height. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Karl Pearson's r for circumference and height is r = Sxy/√(Sxx·Syy) = A. 0.463 B. 0.680 C. 5.008 D. 0.045 Answer: B. 202.8722/√(40.5089 × 2195.7156) = 0.6802. (Key point: Value of r) 2. With n = 9, the t statistic 2.455 is compared with: A. t(0.025, 9) = 2.262 B. z = 1.96 C. t(0.025, 8) = 2.306 D. t(0.025, 7) = 2.365 Answer: D. The test of r uses n − 2 = 7 df; critical value is 2.365, and 2.455 > 2.365 so H0: ρ = 0 is rejected. (Key point: Significance test) 3. r² = 0.463 means: A. About 46.3% of the variation in height is explained by circumference B. 46.3% of trees are tall C. Height rises 46.3% per foot of circumference D. The correlation is 46.3% Answer: A. r² is the proportion of variation in one variable explained by the linear relationship with the other. (Key point: Coefficient of determination) 4. What does significant correlation not prove? A. That the variables move together B. That the relation is positive C. That a larger circumference causes greater height D. That r is between -1 and 1 Answer: C. Correlation shows association only; other factors like age, species, soil may explain it. 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