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The sample number of cases is 27.

The values are provided as \(\sum {{x_i}} = 20,179\) and \(\sum {x_i^2} = 24,657,511\).

Let x represents the sample values.

The sample mean is computed as,

\(\begin{array}{c}\bar x &=& \frac{{\sum {{x_i}} }}{n}\\ &=& \frac{{20179}}{{27}}\\ \approx 747.37\end{array}\)

Thus, the sample mean is 747.37 (in $1000s).

The sample variance is given as,

\({s^2} = \frac{{{S_{xx}}}}{{n - 1}}\)

Where,

\({S_{xx}} = \sum {x_i^2} - \frac{{{{\left( {\sum {{x_i}} } \right)}^2}}}{n}\)

The sample variance is computed as,

\(\begin{array}{c}{s^2} &=& \frac{{{S_{xx}}}}{{n - 1}}\\ &=& \frac{{24657511 - \frac{{{{\left( {20179} \right)}^2}}}{{27}}}}{{27 - 1}}\\ &=& 368320.1652\end{array}\)

Therefore, the sample variance is 368320.1652.

The sample standard deviation is computed as,

\(\begin{array}{c}s &=& \sqrt {{s^2}} \\ &=& \sqrt {368320.1652} \\ &=& 606.89\end{array}\)

Therefore, the sample standard deviation is 606.89(in $1000s).

The maximum possible amount that could be awarded under the two-standard deviation rule is \({\rm{\bar x + 2s}}\) computed as,

\(\begin{array}{c}\bar x + 2s &=& 747.37 + \left( {2*606.89} \right)\\ &=& \$ 1961.16\end{array}\)

Therefore, the maximum possible amount that could be awarded under the two-standard deviation rule is 1961.16 (in $1000s).