Chapter 11.1, Problem 22E

(a)

To determine

To find the Range.

Answer to Problem 22E

Range = $7.7$

Explanation of Solution

Given information: $8.2,10.1,2.6,4.8,2.4,5.6,7.0,3.3$

Formula Used: Range = Highest − Lowest

Calculation:

  $8.2,10.1,2.6,4.8,2.4,5.6,7.0,3.3$

So, In Ascending Order

  $2.4,2.6,3.3,4.8,5.6,7.0,8.2,10.1$

Now,

Range = Highest − Lowest

So,

Range = $10.1-2.4$

Hence,

Range = $7.7$

(b)

To determine

To find the Standard Deviation.

Answer to Problem 22E

Standard Deviation = $2.6$

Explanation of Solution

Given information: $8.2,10.1,2.6,4.8,2.4,5.6,7.0,3.3$

Formula Used: Standard Deviation = $\sqrt{\frac{\sum {\left(\left|x_i-\mu \right|\right)}^2}{N}}$ and Mean $\mu =\frac{Sumofallobservations}{Totalobservations}$

Calculation:

  $8.2,10.1,2.6,4.8,2.4,5.6,7.0,3.3$

Now, in Ascending Order

  $2.4,2.6,3.3,4.8,5.6,7.0,8.2,10.1$

• Mean

Mean = $\frac{Sumofallobservations}{Totalobservations}$

So,

Mean = $\frac{2.4+2.6+3.3+4.8+5.6+7.0+8.2+10.1}{8}$

Now,

Mean = $\frac{44}{8}$

Hence,

Mean $\mu =5.5$

• Standard Deviation

  $x_i=2.4,2.6,3.3,4.8,5.6,7.0,8.2,10.1$

and

Mean $\mu =5.5$

So,

  $\left|x_i-\mu \right|=3.1,2.9,2.2,0.7,0.1,1.5,2.7,4.6$

Now,

  ${\left(\left|x_i-\mu \right|\right)}^2=9.61,8.41,4.84,0.49,0.01,2.25,7.29,21.16$

So,

Standard Deviation = $\sqrt{\frac{\sum {\left(\left|x_i-\mu \right|\right)}^2}{N}}$

Now,

Standard Deviation = $\sqrt{\frac{9.61+8.41+4.84+0.49+0.01+2.25+7.29+21.16}{8}}$

So,

Standard Deviation = $\sqrt{\frac{54.06}{8}}$

Now,

Standard Deviation = $\sqrt{6.76}$

Hence,

Standard Deviation = $2.6$