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Question
Given that the observations are: (9, -4), (10, -3), (11, -1), (12, 0), (13, 1), (14, 3), (15, 5), (16, 8). Find the two lines of regression and estimate the value of y when x = 13·5.
Solution
| x | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | `sumx= 100` |
| y | -4 | -3 | -1 | 0 | 1 | 3 | 5 | 8 | `sum y = 9` |
| `x^2` | 81 | 100 | 121 | 144 | 169 | 196 | 225 | 256 | `sum x^2 = 1292` |
| `y^2` | 16 | 9 | 1 | 0 | 1 | 9 | 25 | 64 | `sum y^2 = 125` |
| xy | -36 | -30 | -11 | 0 | 13 | 42 | 75 | 128 | `sum xy = -77 + 258 = 181` |
`barx = (sumx)/n = 12.5`
`bar y = (sum y)/n = 9/8`
`b_"yx" (sumxy - 1/n sum x sum y)/(sum x^2 - 1/n (sumx)^2)` = `(181 - 1/8 xx 100 xx 9)/(1292 - 1/8 (100)^2)`
`= (181 xx 8 - 900)/(1292 xx - 10000)`
` = (1448 - 900)/336 = 548/336 = 1.631`
`b_"xy" = (sum xy - 1/n sum x sum y)/(sum y^2 - (sumy)^2/n) = (181 - 1/8 xx 100 xx 9)/(125 - 81/8)`
`= 548/919 = 0.596`
Equation of line of y on x is y - 1.125 = 1.63 (x - 12.5)
y - 1.631x = -20.388 + 1.125
y - 1.631x = -19.263
Equation of line x on y is x - 12.5 = 0.596 (y - 1.125)
`x - 0.596y = - 0.6705 + 12.5`
= 11.830
y at x = 13.5
y = -19.263 + 1.631 x 13.5
= 2.7555
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