Let X_1,X_2,…,X_18 be eighteen observations such that ∑i=118X_i-α=36 and ∑i=118Xi-β^2=90, where α and β are distinct real numbers. If the standard…

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Let X_1,X_2,…,X_18 be eighteen observations such that ∑i=118X_i-α=36 and ∑i=118Xi-β^2=90, where α and β are distinct real numbers. If the standard deviation of these observations is 1, then the value of α-β is _______.

MARKS Content Team · Accepted Answer

Correct answer: 4. ∑i=118x_i-α=36,∑i=118xi-β^2=90 ⇒∑i=118x_i=18α+2,∑i=118xi2-2β∑i=118x_i+18β^2=90 Hence ∑xi2=90-18β^2+36βα+2 Given (∑xi2)/18-∑xi18^2=1 ⇒90-18β^2+36βα+2-18α+2^2=18 ⇒5-β^2+2αβ+4β-α^2-4α-4=1 ⇒α-β^2+4α-β=0⇒α-β=0 or 4 As α and β are distinct |α-β|=4

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Let X1,X2,…,X18 be eighteen observations such that ∑i=118Xi-α=36 and ∑i=118Xi-β2=90, where α and β are distinct real numbers. If the standard deviation of these observations is 1, then the value of α-β is _______.

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