Question

For each of the following data sets, compute xbar, s², and s. If appropriate, specify the units in which your answers are expressed.

a.4, 6, 6, 5, 6, 7

b.- \(1, \)4, - \(3, \)0, - \(3, - \)6

c.3/5 %, 4/5 %, 2/5 %, 1/5 %, 1/16 %

d.Calculate the range of each data set in parts a–c.

Short answer

a. mean= 5.66, s² = 1.066, s = 1.03

b. mean = -$1.5, s² = 11.95, s = $3.45c

c. mean = 0.4125%, s² = 0.088, s = 0.29%

d. a = 3, b = $10, c = 0.7375%

Step-by-step solution

Computing xbar, s2, and s for a

$\begin{array}{c}\frac{{\displaystyle \sum \text{χ}}}{\text{n}}\\ \text{=}\frac{\text{4+6+6+5+6+7}}{\text{6}}\\ \text{=}\frac{\text{34}}{\text{6}}\\ \text{=5.66}\end{array}$

$\begin{array}{c}{\text{s}}^{\text{2}}\text{=}\frac{{\displaystyle \sum {\left(\text{χ}-\right)}^2}}{\text{n}-\text{1}}\\ \text{=}\frac{{\left(\text{4}-\text{5.66}\right)}^{\text{2}}\text{+}{\left(\text{6}-\text{5.66}\right)}^{\text{2}}\text{+}{\left(\text{6}-\text{5.66}\right)}^{\text{2}}\text{+}{\left(\text{5}-\text{5.66}\right)}^{\text{2}}\text{+}{\left(\text{6}-\text{5.66}\right)}^{\text{2}}\text{+}{\left(\text{7}-\text{5.66}\right)}^{\text{2}}}{\text{6}-\text{1}}\\ \text{=}\frac{\text{2.7556+0.1156+0.1156+0.4356+0.1156+1.7956}}{\text{5}}\\ \text{=}\frac{\text{5.3336}}{\text{5}}\end{array}$

$\text{=1.066}$

$\begin{array}{c}\text{s=}\sqrt{{\text{s}}^{\text{2}}}\\ \text{=}\sqrt{\text{1.066}}\\ \text{=1.03}\end{array}$

Therefore, mean = 5.66, s² = 1.066, s = 1.03

Calculating xbar, s2, and s for b

$\begin{array}{c}\frac{{\displaystyle \sum \text{χ}}}{\text{n}}\\ \text{=}\frac{-\text{1+4+(}-\text{3)+0+(}-\text{3)+(}-\text{6)}}{\text{6}}\\ \text{=}\frac{-\text{9}}{\text{6}}\\ \text{=}-\text{1.5}\end{array}$

$\begin{array}{c}{\text{s}}^{\text{2}}=\frac{{\displaystyle \sum {\left(\text{χ}-\right)}^2}}{\text{n}-\text{1}}\\ \text{=}\frac{{\left(\left(-\text{1}\right)-\left(-\text{1.5}\right)\right)}^{\text{2}}\text{+}{\left(\text{4}-\left(-\text{1.5}\right)\right)}^{\text{2}}\text{+}{\left(\left(-\text{3}\right)-\left(-\text{1.5}\right)\right)}^{\text{2}}\text{+}{\left(\text{0}-\left(-\text{1.5}\right)\right)}^{\text{2}}\text{+}{\left(\left(-\text{3}\right)-\left(-\text{1.5}\right)\right)}^{\text{2}}\text{+}{\left(\left(-\text{6}\right)-\left(-\text{1.5}\right)\right)}^{\text{2}}}{\text{6}-\text{1}}\\ \text{=}\frac{\text{0.25+30.25+3+3+3+20.25}}{\text{5}}\\ \text{=}\frac{\text{59.75}}{\text{5}}\\ \text{=11.95}\end{array}$

$\begin{array}{c}\text{s=}\sqrt{{\text{s}}^{\text{2}}}\\ \text{=}\sqrt{\text{11.95}}\\ \text{=3.45}\end{array}$

Therefore, mean = -$1.5, s² = 11.95, s = $3.45

Estimating xbar, s2, and s for c

$\begin{array}{c}\frac{{\displaystyle \sum \text{χ}}}{\text{n}}\\ =\frac{0.6+0.8+0.4+0.2+0.0625}{5}\\ =\frac{2.0625}{5}\\ =0.4125\%\end{array}$

$\begin{array}{c}{\text{s}}^{\text{2}}=\frac{{\displaystyle \sum {\left(\text{χ}-\right)}^2}}{\text{n}-\text{1}}\\ =\frac{{\left(0.6-0.4125\right)}^2+{\left(0.8-0.4125\right)}^2+{\left(0.4-0.4125\right)}^2+{\left(0.2-0.4125\right)}^2+{\left(0.0625-0.4125\right)}^2}{5-1}\\ =\frac{0.035+0.15+0.00015+0.045+0.1225}{4}\\ =\frac{0.35}{4}\end{array}$

$\text{=0.088}$

$\begin{array}{c}\text{s=}\sqrt{{\text{s}}^{\text{2}}}\\ \text{=}\sqrt{\text{0.088}}\\ \text{=0.29\%}\end{array}$

Therefore, mean = 0.4125%, s² = 0.088, s = 0.29%

$\begin{array}{c}{\text{s}}^{\text{2}}=\frac{{\displaystyle \sum {\left(\text{χ}-\right)}^2}}{\text{n}-\text{1}}\\ =\frac{{\left(0.6-0.4125\right)}^2+{\left(0.8-0.4125\right)}^2+{\left(0.4-0.4125\right)}^2+{\left(0.2-0.4125\right)}^2+{\left(0.0625-0.4125\right)}^2}{5-1}\\ =\frac{0.035+0.15+0.00015+0.045+0.1225}{4}\\ =\frac{0.35}{4}\end{array}$

Finding the Range

Range = Max – Min

1. 4, 6, 6, 5, 6, 7

$\begin{array}{c}\text{Range}=\text{Max}–\text{Min}\\ =\text{7}–\text{4}\\ =\text{3}\end{array}$

b. - $1, $4, - $3, $0, - $3, - $6

$\begin{array}{c}\text{Range}=\text{Max}–\text{Min}\\ =\text{4}–\left(-\text{6}\right)\\ =\$\text{1}0\end{array}$

c. 3/5 %, 4/5 %, 2/5 %, 1/5 %, 1/16 %

$\begin{array}{c}\text{Range}=\text{Max}–\text{Min}\\ =0.\text{8}–0.0\text{625}\\ =0.\text{7375}\%\end{array}$