- 2070 Chaitra · 4 marks
Regression: chemical yield vs temperature
Short answer
ŷ = 58.581 + 0.1266X. At X=200°C, estimated yield ≈ 83.9%.
Observation on the yield of a chemical reaction taken at various temperatures was recorded as follows:
| X (°C) | 150 | 150 | 200 | 250 | 250 | 300 | 150 |
|---|---|---|---|---|---|---|---|
| Y% | 75.4 | 81.2 | 85.5 | 89 | 90.5 | 96.7 | 75.4 |
| Fit a simple linear regression and estimate value of yield at 200°C. |
Let X = temperature (°C), Y = yield (%), n = 7. Fit Ŷ = a + bX.
Sums (n = 7):
x y xy x² y²
150 75.4 11310 22500 5685.16
150 81.2 12180 22500 6593.44
200 85.5 17100 40000 7310.25
250 89 22250 62500 7921
250 90.5 22625 62500 8190.25
300 96.7 29010 90000 9350.89
150 75.4 11310 22500 5685.16
---------------------------------------------
1450 593.7 125785 322500 50736.15
Σx = 1450, Σy = 593.7, Σxy = 125785, Σx² = 322500, Σy² = 50736.15
Means: x̄ = 207.1429, ȳ = 84.8143
Sxx = Σx² − (Σx)²/n = 22142.8571 Sxy = Σxy − (Σx)(Σy)/n = 2804.2857 Syy = Σy² − (Σy)²/n = 381.9086
Slope b = Sxy/Sxx = 2804.2857/22142.8571 = 0.1266 Intercept a = ȳ − bx̄ = 84.8143 − (0.1266)(207.1429) = 58.5806
Fitted line: Ŷ = 58.581 + 0.1266 X (r = 0.964)
Estimate at X = 200°C:
Ŷ = 58.5806 + 0.12665 × 200 = 58.5806 + 25.3290 = 83.91%
Answer: Ŷ = 58.58 + 0.1266X; the estimated yield at 200°C is about 83.9%.
✓ Answer checked by Claude Sonnet 5.5
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