Let set A consist of 10 elements with a mean of 8 and a variance of 5 . Let set B consist of 20 elements with a mean of 20 and a variance of ^2 . A new set C is formed by adding 2 to each element of set A and subtracting 4 from each element of set B . If the combined variance of the elements in set C is 23 , then the value of ^2 is

Step-by-step solution

Let the new sets formed after transformation be A' and B' . For set A' , the new mean is _ A' = 8 + 2 = 10 . Since adding a constant does not change the variance, the variance of A' is V_ A' = 5 . For set B' , the new mean is _ B' = 20 - 4 = 16 . The variance of B' remains V_ B' = ^2 . The total number of elements in set C is 10 + 20 = 30 . The combined mean of set C is: _C = 10 10 + 20 16 30 = 100 + 320 30 = 14 The formula for the combined variance of two groups is: _C^2 = n₁(V₁ + d₁^2) + n₂(V₂ + d₂^2) n₁ + n₂ whe