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Question

Let X1,X2,...X18 be eighteen observation such that 18i=1(Xiα)=36 and 18i=1(Xiβ)2=90, where α and β are distinct real numbers. If the standard deviation of these observations is 1, then the value of |αβ| is

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Solution

Given, 18i=1(Xiα)=36
xi18α=36
xi=18(α+2)...(1)
Also, 18i=1(Xiβ)2=90
x2i+18β22βxi=90
x2i+18β22β×18(α+2)=90 (using equation (1))
x2i=9018β2+36β(α+2)
Now
σ2=1118x2i(xi18)18=1
(σ=1, given)
118(9018β2+36αβ+72β)(18(α+2)18)2=1
5β2+2αβ+4β(α+2)2=1
5β2+2αβ+4βα244α=1
α2β2+2αβ+4β4α=0
(αβ)(αβ+4)=0
αβ=4
Hence
|αβ|=4,(αβ)

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