Multiple choice

If the mean of a set of observations $\displaystyle x_{1},x_{2},......,x_{10}$ is 20, then the mean of $\displaystyle x_{1}+4,x_{2}+8,x_{3}+12,....,x_{10}+40$ is

  1. $34$
  2. $42$
  3. $38$
  4. $40$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Original mean = (Sum of x_i) / 10 = 20, so Sum = 200. New sum = (x_1+4) + (x_2+8) + ... + (x_10+40) = Sum(x_i) + (4+8+...+40). Sum of arithmetic series = (10/2) * (4+40) = 5 * 44 = 220. New sum = 200 + 220 = 420. New mean = 420 / 10 = 42.

AI explanation

Using the property of arithmetic mean, if each observation is increased by a constant, the mean increases by the average of those constants. The constants added are 4, 8, 12 up to 40 for ten terms, and their sum is 4 times (1 + 2 + ... + 10), which equals 4 times 55, giving 220. The average increase is 220 divided by 10, which is 22; adding this to the original mean of 20 gives a new mean of 42.

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