Question

The standard deviation of $T$is

(a)$22$

(b) $16$

(c)$15.62$

(d)$11.66$

(e)$4$

Short answer

The standard deviation of $T$is option$\left(d\right)$$11.66$

Step-by-step solution

Concept Introduction

The standard deviation of a set of data describes how far values deviate from the mean, with a greater standard deviation indicating a wider range of variance. A low standard deviation implies that the values are close to the mean, whereas a high standard deviation shows that the data are dispersed over a wider range.

Explanation 

Given

${\mu }_X=110$

${\sigma }_X=10$

${\mu }_Y=140$

${\sigma }_Y=12$

Property mean and variance (if $X$and $Y$are independent):

localid="1649858464290" $$\begin{gathered} {\mu }_{aX+bY}=a{\mu }_X+b{\mu }_Y \\ {\sigma }_{aX+bY}^2=a^2{\mu }_X^2+b^2{\mu }_Y^2 \end{gathered}$$

Results we obtain:

localid="1649858468484" $$\begin{gathered} {\mu }_{X+Y/2}={\mu }_X+\frac{1}{2}{\mu }_Y \\ =\left(110+\frac{1}{2}\left(140\right)\right) \\ =180 \\ {\sigma }_{X+Y/2}=\sqrt{{\sigma }_X^2+\frac{1}{2^2}{\sigma }_{X_2}^2} \\ =\sqrt{{10}^2+\frac{1}{4}{12}^2} \\ \approx 11.66. \end{gathered}$$