Consider 10 observation x 1 , x 2 , . .. x 10 , such that ∑ i = 1 10 x i − α = 2 and ∑ i = 1 10 x i − β 2 = 40 , where α , β are positive integers. Let the mean and the variance of the observations be 6 5 and 84 25 respectively. The β α is equal to:

Step-by-step solution

Given observations are x 1 , x 2 , . . . , x 10 ⇒ ∑ i = 1 10 x i − α = 2 ⇒ ∑ i = 1 10 x i − 10 α = 2 Mean, μ = 6 5 = ∑ x i 10 ⇒ ∑ x i = 12 ⇒ 10 α + 2 = 12 ⇒ α = 1 Now, ∑ i = 1 10 x i − β 2 = 40 . Let y i = x i − β ⇒ σ y 2 = 1 10 ∑ y i 2 − y ¯ 2 ⇒ 84 25 = 1 10 ∑ x i − β 2 − ∑ i = 1 10 x i − β 10 2 ⇒ 84 25 = 4 − 12 − 10 β 10 2 ⇒ 6 − 5 β 5 2 = 4 − 84 25 ⇒ 6 − 5 β 5 2 = 16 25 ⇒ 6 − 5 β = ± 4 ⇒ β = 2 5 (not possible) or β = 2 Hence, β α = 2