Question

The arithmetic mean of the data $0,1,2, \ldots \ldots, n$ with frequencies $1,{ }^{n} C_{1},{ }^{n} C_{2}, \ldots,{ }^{n} C_{n}$ is

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

Answer 1

The Correct Option is D

Since, Mean $=\frac{\sum f _{ i } x _{ i }}{\sum f _{ i }}$, where $x _{ i }$ are observations with frequencies $f _{ i }, i =1,2, \ldots . n$
The required mean is given by,
$\bar{X}=\frac{0.1+1 \cdot{ }^{n} C_{1}+2 \cdot{ }^{n} C_{2}+\ldots .+n \cdot{ }^{n} C_{n}}{1+{ }^{n} C_{1}+{ }^{n} C_{2}+\ldots +{ }^{n} C_{n}}$
$=\frac{\displaystyle\sum_{r=0}^{n} n \cdot{ }^{n} C_{r}}{\displaystyle\sum_{r=0}^{n}{ }^{n} C_{r}}=\frac{n \displaystyle\sum_{r=0}^{n}{ }^{n-1} C_{r-1}}{\displaystyle\sum_{r=0}^{n}{ }^{n} C_{r}}$
$=\frac{n 2^{n-1}}{2^{n}}=\frac{n}{2}$