146 192 174 177 159 87 178 152 144 148

Sol (a)

\[\begin{array}{|c|c|} \hline X&X-\overline{X}&(X-\overline{X})^{2}\\ \hline 146&-9.7000&94.0900\\ \hline 192&36.3000&1317.6900\\ \hline 174&18.3000&334.8900\\ \hline 177&21.3000&453.6900\\ \hline 159&3.3000&10.8900\\ \hline 87&-68.7000&4719.6900\\ \hline 178&22.3000&497.2900\\ \hline 152&-3.7000&13.6900\\ \hline 144&-11.7000&136.8900\\ \hline 148&-7.7000&59.2900\\ \hline \sum x=1557&&\sum (x-\overline{x})^{2}=638.1000\\ \hline \end{array}\]

sample Size \(\displaystyle={n}={10}\)

Mean \(\displaystyle=\overline{{{X}}}={\frac{{\sum{x}}}{{{n}}}}={155.7}\)

Std dev. \(\displaystyle={s}=\sqrt{{{\frac{{\sum{\left({x}-\overline{{{x}}}\right)}^{{{2}}}}}{{{n}-{1}}}}}}={29.1}\)

\[\begin{array}{|c|c|} \hline \text{mean}&155.7\ lbs\\ \hline \text{standard deviation}&29.1\ lbs\\ \hline \end{array}\]

Sol (b)

one standard deviation range: \(\displaystyle{\left({155.7}-{29.1},{155.7}+{29.1}\right)}={\left({126.6},{184.8}\right)}\)

Data values in this range: [146 174 177 159 178 152 144 148]: 8 values in this range NK Percentage \(\displaystyle={\frac{{{8}}}{{{10}}}}={0.8}\)

80% of the data lies within one standard deviation of the mean

Sol (C)

two standard deviation range: \(\displaystyle{\left({155.7}-{2}\cdot{29.1},{155.7}+{2}\cdot{29.1}\right)}={\left({97.4},{214}\right)}\)

Data values in this range: [146 174 177 159 178 152 144 148 192]: 9 values in this range

Percentage \(\displaystyle={\frac{{{9}}}{{{10}}}}={0.9}\)

90 % of the data lies within two standard deviation of the mean