Question

Question 14
If each observation of the data is increased by 5, then their mean:

A) Remains the same
B) Becomes 5 times the original mean
C) Is decreased by 5
D) Is increases by 5

Answer 1

The answer is D.
Let $x_1,x_2,\dots \dots x_n$ be the n observations.
Then, old mean ${\overline{x}}_{old}=\frac{{\sum }_{i=1}^n{x}_i}{n}.....\left(i\right)$
Now, adding 5 in each observation, the new mean becomes
${\overline{x}}_{new}=\frac{(x_1+5)+(x_2+5)+\dots \dots +(x_n+5)}{n}$
$\Rightarrow {\overline{x}}_{new}=\frac{(x_1+x_2+\dots \dots x_n)+5n}{n}$
$\Rightarrow {\overline{x}}_{new}=\frac{{\sum }_{i=1}^n{x}_i}{n}+5={\overline{x}}_{old}+5\left[from\right(i\left)\right]$
$\Rightarrow {\overline{x}}_{new}={\overline{x}}_{old}+5$
Hence, the new mean is increased by 5.