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प्रश्न
The equations of two regression lines are
2x + 3y − 6 = 0
and 3x + 2y − 12 = 0 Find
- Correlation coefficient
- `sigma_"X"/sigma_"Y"`
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उत्तर
The given regression equations are
2x + 3y – 6 = 0 and 3x + 2y − 12 = 0
(i) 2x + 3y = 6
3y = – 2x + 6
`y = (– 2)/3x + 2`
`b_( yx) = (-2)/3`
3x + 2y = 12
3x = – 2y = 12
`x = (-2)/3y + 4`
`b_(xy) = (-2)/3`
`b_( yx).b_(xy) = (-2)/3 xx (-2)/3 = 4/9 ∈ [0, 1]`
∴ Our assumption is correct.
∴ `r^2 = b_( yx).b_(xy)`
`r^2 = 4/9`
`r = ±2/3`
Since `b_( yx)` and `b_(xy)` are negative ∴ r =`(-2)/3`
(ii) `b_(xy) = (r . sigma_y)/sigma_x`
`(-2)/3 = (-2)/3 . sigma_x/sigma_y`
∴ `sigma_x/sigma_y = 1`
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| x | y | `x - barx` | `y - bary` | `(x - barx)(y - bary)` | `(x - barx)^2` | `(y - bary)^2` |
| 1 | 5 | – 2 | – 4 | 8 | 4 | 16 |
| 2 | 7 | – 1 | – 2 | `square` | 1 | 4 |
| 3 | 9 | 0 | 0 | 0 | 0 | 0 |
| 4 | 11 | 1 | 2 | 2 | 4 | 4 |
| 5 | 13 | 2 | 4 | 8 | 1 | 16 |
| Total = 15 | Total = 45 | Total = 0 | Total = 0 | Total = `square` | Total = 10 | Total = 40 |
Mean of x = `barx = square`
Mean of y = `bary = square`
bxy = `square/square`
byx = `square/square`
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∴ Regression equation x on y is `square`
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∴ Regression equation of y on x is `square`
Mean of x = 53
Mean of y = 28
Regression coefficient of y on x = – 1.2
Regression coefficient of x on y = – 0.3
a. r = `square`
b. When x = 50,
`y - square = square (50 - square)`
∴ y = `square`
c. When y = 25,
`x - square = square (25 - square)`
∴ x = `square`
| x | y | xy | x2 | y2 |
| 6 | 9 | 54 | 36 | 81 |
| 2 | 11 | 22 | 4 | 121 |
| 10 | 5 | 50 | 100 | 25 |
| 4 | 8 | 32 | 16 | 64 |
| 8 | 7 | `square` | 64 | 49 |
| Total = 30 | Total = 40 | Total = `square` | Total = 220 | Total = `square` |
bxy = `square/square`
byx = `square/square`
∴ Regression equation of x on y is `square`
∴ Regression equation of y on x is `square`
