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Let X = 11 , 12 , 13 , … . , 40 , 41 and Y = 61 , 62 , 63 , . . . , 90 , 91 be the two sets of observations. If x ¯ and y ¯ are their respective means and σ 2 is the variance of all the observations in X ∪ Y , then x ¯ + y ¯ - σ 2 is equal to ________

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Step-by-step solution

We have, X = 11 , 12 , 13 , … . , 40 , 41 ⇒ n X = 31 Y = 61 , 62 , 63 , . . . , 90 , 91 ⇒ n Y = 31 So, x ¯ = ∑ i = 11 41 i 31 = 1 31 41 × 42 2 - 10 × 11 2 = 26 y ¯ = ∑ i = 61 91 i 31 = 1 31 91 × 92 2 - 60 × 61 2 = 73 Combined mean, μ = 31 × 26 + 31 × 76 31 + 31 ⇒ μ = 26 + 76 2 = 51 Also, σ 2 = 1 62 × ∑ i = 1 31 x i - μ 2 + ∑ i = 1 31 y i - μ 2 Since, x i ∈ X are in A.P. with 31 elements common

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