Let X 1 , X 2 , … , X 18 be eighteen observations such that ∑ i = 1 18 X i - α = 36 and ∑ i = 1 18 X i - 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 _______.

Step-by-step solution

∑ i = 1 18 x i - α = 36 , ∑ i = 1 18 x i - β 2 = 90 ⇒ ∑ i = 1 18 x i = 18 α + 2 , ∑ i = 1 18 x i 2 - 2 β ∑ i = 1 18 x i + 18 β 2 = 90 Hence ∑ x i 2 = 90 - 18 β 2 + 36 β α + 2 Given ∑ x i 2 18 - ∑ x i 18 2 = 1 ⇒ 90 - 18 β 2 + 36 β α + 2 - 18 α + 2 2 = 18 ⇒ 5 - β 2 + 2 α β + 4 β - α 2 - 4 α - 4 = 1 ⇒ α - β 2 + 4 α - β = 0 ⇒ α -