If the mean of a set of observations x1, x2, x3, ..., x10 is 20, then the mean of x1 + 5, x2 + 10, x3 + 15, ..., x10 + 50 is :
47·5
The mean of n observations equals their sum divided by n. If different amounts are added to the observations, the new sum equals the old sum plus the total of the amounts added.
Step 1 — Sum of the original observations: \(x_1 + x_2 + \dots + x_{10} = 10 \times 20 = 200\).
Step 2 — The amounts added are 5, 10, 15, ..., 50, i.e. 5 times 1 to 10: \(5(1 + 2 + \dots + 10) = 5 \times 55 = 275\).
Step 3 — New sum = 200 + 275 = 475, so the new mean = \(\frac{475}{10} = 47.5\). Equivalently, the mean rises by the mean of the added amounts, 27·5, giving 20 + 27·5.
The values 53·5, 41·5 and 37·5 do not match this result, so the remaining choice does not apply either.
47·5 option is correct.
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