3. Example-3 (Class-X & Class-Y)
Find Regression line equations from the following data
Class-Y
Class-X | 40 - 49 | 50 - 59 | 60 - 69 | 70 - 79 | 80 - 89 | 90 - 99 | | 40 - 49 | 3 | 5 | 4 | 0 | 0 | 0 | | 50 - 59 | 2 | 6 | 6 | 2 | 0 | 0 | | 60 - 69 | 1 | 4 | 10 | 5 | 2 | 0 | | 70 - 79 | 0 | 0 | 5 | 10 | 8 | 1 | | 80 - 89 | 0 | 0 | 1 | 4 | 6 | 5 | | 90 - 99 | 0 | 0 | 0 | 2 | 4 | 4 |
Solution:
| | | C.I.`(y)` | 40 - 49 | 50 - 59 | 60 - 69 | 70 - 79 | 80 - 89 | 90 - 99 | | | | | | | | M.V.`(y)` | 44.5 | 54.5 | 64.5 | 74.5 | 84.5 | 94.5 | | | | | | C.I.`(x)` | M.V.`(x)` | `dy` `dx` | -2 | -1 | 0 | 1 | 2 | 3 | `f_x` | `fdx` | `fdx^2` | `fdxdy` | | 40 - 49 | 44.5 | -2 | [12]3 | [10]5 | [0]4 | [0]0 | [0]0 | [0]0 | 12 | -24 | 48 | 22 | | 50 - 59 | 54.5 | -1 | [4]2 | [6]6 | [0]6 | [-2]2 | [0]0 | [0]0 | 16 | -16 | 16 | 8 | | 60 - 69 | 64.5 | 0 | [0]1 | [0]4 | [0]10 | [0]5 | [0]2 | [0]0 | 22 | 0 | 0 | 0 | | 70 - 79 | 74.5 | 1 | [0]0 | [0]0 | [0]5 | [10]10 | [16]8 | [3]1 | 24 | 24 | 24 | 29 | | 80 - 89 | 84.5 | 2 | [0]0 | [0]0 | [0]1 | [8]4 | [24]6 | [30]5 | 16 | 32 | 64 | 62 | | 90 - 99 | 94.5 | 3 | [0]0 | [0]0 | [0]0 | [6]2 | [24]4 | [36]4 | 10 | 30 | 90 | 66 | | | | `f_y` | 6 | 15 | 26 | 23 | 20 | 10 | 100 | 46 | 242 | 187 | | | | `fdy` | -12 | -15 | 0 | 23 | 40 | 30 | 66 | | | | | | | `fdy^2` | 24 | 15 | 0 | 23 | 80 | 90 | 232 | | | | | | | `fdxdy` | 16 | 16 | 0 | 22 | 64 | 69 | 187 | | | |
`bar x = A + (sum fdx)/n * h_x`
`=64.5 + 46/100 * 10`
`=64.5 + 0.46 * 10`
`=64.5 + 4.6`
`=69.1`
`bar y = B + (sum fdy)/n * h_y`
`=64.5 + 66/100 * 10`
`=64.5 + 0.66 * 10`
`=64.5 + 6.6`
`=71.1`
`byx = (n * sum fdxdy - sum fdx * sum fdy ) / (n * sum fdx^2 - (sum fdx)^2) * (h_y)/(h_x)`
`=(100 * 187 - 46 * 66 )/(100 * 242 - (46)^2) * 10/10`
`=(18700 - 3036 )/(24200 - 2116) * 10/10`
`=15664/22084 * 10/10`
`=0.7093`
Regression Line y on x
`y - bar y = byx (x - bar x)`
`y - 71.1 = 0.7093 (x - 69.1)`
`y - 71.1 = 0.7093 x - 49.0121`
`y = 0.7093 x - 49.0121 + 71.1`
`y = 0.7093 x + 22.0879`
`bxy = (n * sum fdxdy - sum fdx * sum fdy ) / (n * sum fdy^2 - (sum fdy)^2) * (h_x)/(h_y)`
`=(100 * 187 - 46 * 66 )/(100 * 232 - (66)^2) * 10/10`
`=(18700 - 3036 )/(23200 - 4356) * 10/10`
`=15664/18844 * 10/10`
`=0.8312`
Regression Line x on y
`x - bar x = bxy (y - bar y)`
`x - 69.1 = 0.8312 (y - 71.1)`
`x - 69.1 = 0.8312 y - 59.1016`
`x = 0.8312 y - 59.1016 + 69.1`
`x = 0.8312 y + 9.9984`
This material is intended as a summary. Use your textbook for detail explanation.
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