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The mean of n observations is X ¯ . If the first observation is increased by 1 2 , second by 2 2 and so on, then the new mean is

Options

  1. AX ¯ + n n + 1 2 n + 1 6
  2. BX ¯ + n + 1 2 n + 1 6
  3. CX ¯ + n + 1 2
  4. DX ¯ + n + 1 4

Correct answer

B. X ¯ + n + 1 2 n + 1 6

Step-by-step solution

Let, x 1 , x 2 , . . . . . x n be n observations. Then, X ¯ = 1 n Σ x i . Let, y 1 = x 1 + 1 2 , y 2 = x 2 + 2 2 , y 3 = x 3 + 3 2 , . . . . , y n = x n + n 2 Then, the mean of the new series is 1 n Σ y i = 1 n Σ x i + i 2 = 1 n Σ x i + 1 n 1 2 + 2 2 + 3 2 + . . . . + n 2 = X ¯ + 1 n ⋅ n n + 1 2 n + 1 6 = X ¯ + n + 1 2 n + 1 6

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