- 2065 Chaitra · 6 marks
Regression equation for demand of fish
Short answer
The regression line is Q = 110.20 − 0.743P, fitted by least squares. For a price of Rs 60 per kg, forecast demand is 65.6 tonnes.
The following data shows the demands for fish when the prices are as shown. Calculate the hypothesized regression equation. What shall be the demand if the price is set to be Rs. 60 per kg?
| S.N | Price per kg. | Quantity (tones) |
|---|---|---|
| 1 | 64 | 65 |
| 2 | 53 | 75 |
| 3 | 67 | 56 |
| 4 | 52 | 69 |
| 5 | 82 | 51 |
| 6 | 59 | 65 |
| 7 | 67 | 63 |
| 8 | 71 | 55 |
Method
Fit the hypothesised linear demand Q = a + bP by least squares, where P = price (Rs/kg) and Q = quantity (tonnes). n = 8. b = (nΣPQ − ΣPΣQ) / (nΣP² − (ΣP)²), a = (ΣQ − bΣP) / n
| S.N | P | Q | PQ | P² |
|---|---|---|---|---|
| 1 | 64 | 65 | 4,160 | 4,096 |
| 2 | 53 | 75 | 3,975 | 2,809 |
| 3 | 67 | 56 | 3,752 | 4,489 |
| 4 | 52 | 69 | 3,588 | 2,704 |
| 5 | 82 | 51 | 4,182 | 6,724 |
| 6 | 59 | 65 | 3,835 | 3,481 |
| 7 | 67 | 63 | 4,221 | 4,489 |
| 8 | 71 | 55 | 3,905 | 5,041 |
| Σ | 515 | 499 | 31,618 | 33,833 |
b = (8 × 31,618 − 515 × 499) / (8 × 33,833 − 515²) = (252,944 − 256,985) / (270,664 − 265,225) = −4,041 / 5,439 = −0.7430
a = (499 − (−0.7430)(515)) / 8 = (499 + 382.65) / 8 = 110.20
Regression equation: Q = 110.20 − 0.743 P
Demand at Rs 60 per kg
Q = 110.20 − 0.743 × 60 = 110.20 − 44.58 = 65.6 tonnes
Answer: Q = 110.20 − 0.743P; at Rs 60/kg demand is about 65.6 tonnes. The negative slope means demand falls by about 0.74 tonne for every Rs 1 rise in price, as expected from the law of demand.
✓ Answer checked by Claude Opus 5.5 (high effort)
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