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The mean and standard deviation of 10 observations x 1 , x 2 , x 3 . . . . . x 10 are x ¯ and σ respectively. Let 10 is added to x 1 , x 2 . . . . . x 9 and 90 is subtracted from x 10 . If still, the standard deviation is the same, then x 10 - x ¯ is equal to

Options

  1. A35
  2. B45
  3. C55
  4. D50

Correct answer

B. 45

Step-by-step solution

∑ i = 1 10 x i - x ¯ 2 10 = σ 2 … 1 ∑ i = 1 9 x i + 10 - x ¯ 2 + x 10 - 90 - x ¯ 2 10 = σ 2 … 2 From 1 and ( 2 ) ∑ i = 1 10 x i - x ¯ 2 10 = ∑ i = 1 10 x i - x ¯ 2 + 20 ∑ i = 1 9 x i - x ¯ - 180 x 10 - x ¯ + 100 × 9 + 8100 10 2 × ∑ i = 1 9 x i - x ¯ + 2 x 10 - x ¯ - 20 x 10 - x ¯ + 90 + 810 = 0 ⇒ 900 = 20 x 10 - x ¯ ⇒ x 10 - x ¯ = 4 5

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