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Question
The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.
Solution
Let those two numbers be x and y.
∴ `overline x = 9 = (6 + 7 + 10 + 12 + 12 + 13 + x + y)/8`
or 72 = 60 + x + y
∴ x + y = 12 ...(i)
Variance σ2 = `1/n^2 [nsumx_i^2 - (sumx_i)^2]`
∴ `sumx _i = 9`
∴ `9.25 = 1/64 [8 xx sumx_i^2 - (sumx_i)^2]`
∴ `sumx_i = 72`
`8 xx sumx_i = 9.25 xx 64 + 72 xx 72`
= 592 + 5184
= 5776
∴ `sumx_i^2 = 5776/8`
= 722
or `sumx_i^2 = 722`
= 62 + 72 + 102 + 122 + 122 + 132 + x2 + y2
722 = 36 + 49 + 100 + 144 + 144 + 169 + x2 + y2
= 642 + x2 + y2
x2 + y2
= 722 – 642
= 80
∴ x2 + y2 = 80 ....(ii)
From equations (i) and (ii)
or x2 + (12 – x)2 = 80
or 2x2 – 24x + 144 = 80
or x2 – 12x + 32 = 0
(x – 4)(x – 8) = 0
∴ x = 4 or 8
∴ y = 8 or 4
Hence, those two numbers are 4 and 8.
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