Chapter 6.1, Problem 15E

To determine

To find: the standard deviation of X and explain it

Answer to Problem 15E

  $\sigma =2.4618$

Explanation of Solution

Given:

Score(X)	1	2	3	4	5	6	7	8	9
Probability	0301	0.176	0.125	0.097	0.079	0.067	0.058	0.051	0.046

Formula used:

  $\sigma =\sqrt{E\left(X^2\right)+{\left\{E\left(X\right)\right\}}^2}$

Calculation:

$x$	$\text{P}\left(x\right)$	$x\text{P}\left(x\right)$	$x^2$	$x^2\text{P}\left(x\right)$
1	0.301	0.301	1	0.301
2	0.176	0.352	4	0.704
3	0.125	0.375	9	1.125
4	0.097	0.388	16	1.552
5	0.079	0.395	25	1.975
6	0.067	0.402	36	2.412
7	0.058	0.406	49	2.842
8	0.051	0.408	64	3.264
9	0.046	0.414	81	3.726
		3.441		17.901

  $\begin{array}{c}E\left(X\right)={\displaystyle \sum xP(x)}\\ =3.441\\ E\left(X^2\right)=x^{2}P\left(x\right)\\ =17.901\end{array}$

  $\begin{array}{c}\sigma =\sqrt{17.901-{3.441}^2}\\ =\sqrt{17.901-11.8405}\\ =\sqrt{6.0605}\\ =2.4618\end{array}$

Standard deviation is 2.4618 and mean is 3.441 therefore the scores on an arbitrary Choose varies 2.4618 from the mean days 3.441