Question

The arithmetic mean the data $0,1,2,,....n$ with frequencies $1,{{}^{n}C}_1,{{}^{n}C}_2,....{{}^{n}C}_n$ is

• A. $n$
• B. $\frac{2^n}{n}$
• C. $n+1$
• D. $\frac{n}{2}$

Answer 1

Answer: (D)

$\frac{n}{2}$

Since, $\text{Mean}=\frac{\sum f_i{x}_i}{\sum f_i}$, where $x_i$ are observations with frequencies $f_i,i=1,2,....n$
The required mean is given by,
$\overline{X}=\frac{0.1+1.{{}^{n}C}_1+2.{{}^{n}C}_2+....+n.{{}^{n}C}_n}{1+{{}^{n}C}_1+{{}^{n}C}_2+....+{{}^{n}C}_n}$
$=\frac{{\sum }_{r=0}^{n}n.{{}^{n}C}_r}{{\sum }_{r=0}^n{{}^{n}C}_r}=\frac{n{\sum }_{r=0}^n.{{}^{n-1}C}_{r-1}}{{\sum }_{r=0}^n{{}^{n}C}_r}=\frac{n{2}^{n-1}}{2^n}=\frac{n}{2}$