Question

In 1997 a woman sued a computer keyboard manufacturer,charging that her repetitive stress injuries werecaused by the keyboard (Genessy v. Digital EquipmentCorp.).The injury awarded about \(3.5 million for painand suffering, but the court then set aside that awardas being unreasonable compensation. In making this determination, the court identified a “normative” group of27 similar cases and specified areasonable award as onewithin two standard deviations of the mean of the awardsin the 27 cases. The 27 awards were (in \)1000s) 37, 60,75, 115, 135, 140, 149, 150, 238, 290, 340, 410, 600, 750,750, 750, 1050, 1100, 1139, 1150, 1200, 1200, 1250,1576, 1700, 1825, and 2000, from which\(\sum {{x_i} = } \)20,179,\(\sum {x_i^2} = 24,657,511\). What is the maximum possible amount that could be awarded under the two- standard deviation rule?

Short answer

The maximum possibleamount that could be awarded under the two-standard deviationrule is $1961.16(in $1000s).

Step-by-step solution

Given information

The sample number of cases is 27.

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

Compute the maximum possible amount

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).

Compute the maximum possible amount

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).