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There are two sets of data, A and B , containing 20 and 30 observations respectively. The mean and variance of set A are 12 and 11 respectively. When all 50 observations are combined, the overall mean is 15 and the overall variance is 20 . If the mean and variance of set B are and ^2 respectively, then the value of + ^2 is equal to

Options

  1. A33
  2. B47
  3. C43
  4. D31

Correct answer

A. 33

Step-by-step solution

Let the sum of observations in set A be x_A and in set B be x_B . For set A : n_A = 20 , x _A = 12 , _A^2 = 11 x_A = n_A x _A = 20 12 = 240 x_A^2 = n_A ( _A^2 + x _A^2) = 20 (11 + 144) = 20 155 = 3100 For the combined dataset: N = 50 , x _ comb = 15 , _ comb ^2 = 20 x_ comb = 50 15 = 750 x_ comb ^2 = 50 ( _ comb ^2 + x _ comb ^2) = 50 (20 + 225) = 50 245 = 12250 For set B : n_B = 30 x_B = x_ comb - x_A = 750 - 240 = 510 Mean of set B , = x_B n_B = 510 30 = 17 x_B^2 = x_ comb ^2 - x_A^2 = 12250 - 3100 = 9150 Varianc

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