Question

A class has $15$ students whose ages are $14,17,15,14,21,17,19,20,16,18,20,17,16,19$ and $20$ years. One student is selected in such a manner that each has the same chance of being chosen and the age $X$ of the selected student is recorded. Then standard deviation is:

• A. $1.16$
• B. $1.74$
• C. $2.21$
• D. $1.2$

Answer 1

The correct option is C $2.21$

Given the class of $15$ students with their ages.
Form the given data we can draw a table
$\begin{array}{ccccccccc}\text{X}& \text{14}& \text{15}& \text{16}& \text{17}& \text{18}& \text{19}& \text{20}& \text{21}\\ \text{f(x)}& \text{2}& \text{1}& \text{2}& \text{3}& \text{1}& \text{2}& \text{3}& \text{1}\end{array}$
$P(X=14)=2/15$
$P(X=15)=1/15$
$P(X=16)=2/15$
$P(X=17)=3/15$
$P(X=18)=1/15$
$P(X=19)=2/15$
$P(X=20)=3/15$
$P(X=21)=1/15$
Hence, the required probability distribution is,
$\begin{array}{ccccccccc}\text{X}& \text{14}& \text{15}& \text{16}& \text{17}& \text{18}& \text{19}& \text{20}& \text{21}\\ \text{P(x)}& \text{2/15}& \text{1/15}& \text{2/15}& \text{3/15}& \text{1/15}& \text{2/15}& \text{3/15}& \text{1/15}\end{array}$
Therefore $E\left(X\right)$ is
$E\left(X\right)=\sum _{i=1}^n{x}_i{p}_i$
$=14\times \frac{2}{15}+15\times \frac{1}{15}+16\times \frac{2}{15}+17\times \frac{3}{15}+18\times \frac{1}{15}+19\times \frac{2}{15}+20\times \frac{3}{15}+21\times \frac{1}{15}$
$=\frac{28+15+32+51+18+38+60+21}{15}=\frac{263}{15}$
$\Rightarrow E\left(X\right)=17.53$
And $E\left(X^2\right)$ is:
$E\left(X^2\right)=\sum _{i=1}^n{x}_i^2.p\left(x_i\right)$
$=(14{)}^2\times \frac{2}{15}+(15{)}^2\times \frac{1}{15}+(16{)}^2\times \frac{2}{15}+(17{)}^2\times \frac{3}{15}+(18{)}^2\times \frac{1}{15}+(19{)}^2\times \frac{2}{15}+(20{)}^2\times \frac{3}{15}+(21{)}^2\times \frac{1}{15}$
$=\frac{392+225+512+867+324+722+1200+441}{15}=\frac{4683}{15}$
$\Rightarrow E\left(X^2\right)=312.2$
Then Variance, $Var\left(X\right)=E\left(X^2\right)-\left(E\right(X){)}^2=312.2-(17.53{)}^{2}312.2-307.3009=4.899$
And standard deviation $=\sqrt{Var\left(X\right)}=\sqrt{4.899}$
$\Rightarrow$ Standard deviation $=2.21$