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Let x ¯ , M and σ 2 be respectively the mean, mode and variance of n observations x 1 ,   x 2 , . ... ,   x n and d i = - x i - a ,   i = 1 ,   2 , . ... ,   n , where a is any number. Statement I: Variance of d 1 ,   d 2 , . . . ,   d n is σ 2 . Statement II: Mean and mode of d 1 ,   d 2 , . . . . ,   d n are - x ¯ - a and - M - a , respectively.

Options

  1. AStatement I and Statement II are both true
  2. BStatement I and Statement II are both false
  3. CStatement I is true and Statement II is false
  4. DStatement I is false and Statement II is true

Correct answer

A. Statement I and Statement II are both true

Step-by-step solution

Mean of d 1 ,   d 2 ,   d 3 . . .   d n = ∑     - x i - a   n   = - x → - a And, the mode of d 1 ,   d 2 ,   d 3 . . .   d n will be - M - a Also, we know that the variance σ 2 is unaffected by change of origin, hence it remains same.

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