Multiple choice

In a series of 2n observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant b in each of these observations, the mean and standard deviation of the new set become 5 and 20, respectively. Then the value of a2 + b2 is equal to:

  1. 425

  2. 650

  3. 250

  4. 925
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A Correct answer
Explanation

Observations are n of 'a' and n of '-a'. Mean = (na - na) / 2n = 0. Adding b makes mean = b = 5. Variance of original set = (n*a^2 + n*(-a)^2) / 2n = a^2. Standard deviation = 20, so variance = 400. Thus a^2 = 400. Then a^2 + b^2 = 400 + 25 = 425.

AI explanation

The original mean of the 2n observations of a and negative a is 0, so adding a constant b to each observation changes the mean to b, meaning b is 5. The original standard deviation of a and negative a is a, and adding a constant does not change the standard deviation, so a equals 20. The sum of their squares is 20 squared plus 5 squared, which equals 425.

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