--- title: "Strength of antibiotics-bacteria relationship" subject: "Probability and Statistics" chapter: 5 key: "semester_5/probability_and_statistics:antibiotics-bacteria-correlation" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/antibiotics-bacteria-correlation site: BCT Engineering --- # Strength of antibiotics-bacteria relationship Probability and Statistics · Chapter 5: Correlation and regression - Asked in: 2074 Chaitra, 4 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/antibiotics-bacteria-correlation - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5 ## Question In trying to evaluate the effectiveness of antibiotics in killing bacteria, a research institute compiled the following information | Antibiotics (mg) | 12 | 15 | 14 | 16 | 17 | 10 | |---|---|---|---|---|---|---| | Bacteria | 5 | 7 | 5.6 | 7.2 | 8.6 | 6.2 | Find strength and direction of relationship between them. ## Short answer Karl Pearson's r is 0.7408. The positive sign shows a direct relationship: bacteria count increases as antibiotic dose increases. This indicates a moderately strong to strong linear relationship, with r squared 0.549. ## Answer The strength and direction of a linear relationship are measured by Karl Pearson's correlation coefficient r. Let x = antibiotics (mg), y = bacteria. Sums (n = 6): ```text x y xy x² y² 12 5 60 144 25 15 7 105 225 49 14 5.6 78.4 196 31.36 16 7.2 115.2 256 51.84 17 8.6 146.2 289 73.96 10 6.2 62 100 38.44 --------------------------------------------- 84 39.6 566.8 1210 269.6 ``` Σx = 84, Σy = 39.6, Σxy = 566.8, Σx² = 1210, Σy² = 269.6 Means: x̄ = 14.0000, ȳ = 6.6000 Sxx = Σx² − (Σx)²/n = 34.0000 Sxy = Σxy − (Σx)(Σy)/n = 12.4000 Syy = Σy² − (Σy)²/n = 8.2400 Slope b = Sxy/Sxx = 12.4000/34.0000 = 0.3647 Intercept a = ȳ − bx̄ = 6.6000 − (0.3647)(14.0000) = 1.4941 r = Sxy/√(Sxx·Syy) = 12.4000/√(34.0000 × 8.2400) = 12.4000/16.7380 = **0.7408** **Direction:** r is positive, so the relationship is direct: as the antibiotic dose increases, the bacteria measure also increases. **Strength:** r = 0.741 indicates a moderately strong to strong positive linear relationship (r² = 0.549, so 54.9% of the variation in bacteria is explained by the antibiotic amount; the rest is due to other factors). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Strength and direction of linear relationship are measured by: A. The regression intercept B. Karl Pearson's correlation coefficient r C. The standard error D. The mean Answer: B. r gives strength (magnitude) and direction (sign). (Key point: Measure used) 2. With Sxy = 12.4, Sxx = 34, Syy = 8.24, r equals: A. 0.365 B. 0.549 C. 0.741 D. 1.5 Answer: C. 12.4/√(34 × 8.24) = 12.4/16.738 = 0.7408. (Key point: Value of r) 3. r is positive. This means: A. Bacteria fall as antibiotic dose rises B. No relationship C. Bacteria is constant D. Bacteria measure rises as antibiotic dose rises (direct relationship) Answer: D. Positive r means the variables move in the same direction. (Key point: Direction) 4. r² = 0.549 means: A. 54.9% of variation in bacteria is explained by the antibiotic amount B. 54.9% of bacteria are killed C. r is 54.9% D. Antibiotic explains 45.1% of variation Answer: A. r² is the proportion of variation explained; the rest is due to other factors. 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