Calculate Population cov(x,y), Sample cov(x,y) 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 | | | |
Population Cov(x,y) :
Population `Cov(x,y) = (sum fdxdy - (sum fdx * sum fdy)/n)/(n)`
`=(187 - (46 xx 66)/100)/100`
`=(187 - (3036)/100)/100`
`=(187 - 30.36)/100`
`=(156.64)/100`
`=1.5664`
Sample Cov(x,y) :
Sample `Cov(x,y) = (sum fdxdy - (sum fdx * sum fdy)/n)/(n-1)`
`=(187 - (46 xx 66)/100)/99`
`=(187 - (3036)/100)/99`
`=(187 - 30.36)/99`
`=(156.64)/99`
`=1.5822`
This material is intended as a summary. Use your textbook for detail explanation.
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