Question

A woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard. The injury awarded about $3.5 million for pain and suffering, but the court then set aside that award as being unreasonable compensation. In making this determination, the court identified a "normative" group of 27 similar cases and specified a reasonable award as one within two standard deviations of the mean of the awards in the 27 cases. The 27 awards were (in $1000s):
37, 60, 72, 112, 138, 145, 147, 150, 238, 290, 340, 410, 600, 750, 750, 751, 1050, 1100, 1137, 1150, 1200, 1200, 1250, 1579, 1700, 1825, and 2000.
From which:

Σxᵢ = 20,180 and Σxᵢ² = 23,465,528

What is the maximum possible amount that could be awarded under the two- standard deviation rule?

Answer 1

Answer:

The maximum possible amount that could be awarded under the two-deviation rule is $ 1.883 Million.

Step-by-step explanation:

In order to find the maximum possible amount of the award, the formula is given as

                                                      $x=\bar{x}+2s$

Here

$\bar{x}$ is the mean of the samples and is given as

                                                   $\bar{x}= \frac{\sum_{i=1}^{n} x_i}{n}$

s is the standard deviation which is given as

                                $s=\sqrt{\frac{1}{n-1} (\sum x_i^2-\frac{1}{n} (\sum x_i)^2 )}$

Putting values as

                             $\sum x_i^2=23,465,528 \ (1000s) \\ \sum x_i=20,180 \ (1000s) \\ n=27$

Substituting the values in the equation of mean is as

                              $\bar{x}= \frac{20,180}{27} \\ \bar{x}= 747.407 \ (1000s)$

Where as standard deviation is given as

                     $s=\sqrt{\frac{1}{27-1} (23,465,528-\frac{1}{27} (20,180)^2 )} \\ s=\sqrt{\frac{1}{26} (23,465,528-15082681.4815 )} \\s=\sqrt{\frac{1}{26} (8382846.51)} \\s=\sqrt{322417.17} \\s=567.81 \ (1000s)$

Putting this in the equation of maximum amount gives

                             $x=\bar{x}+2.s \\ x=747.407 +2(567.81) \\x=1883.04 \ (1000s)$

So the maximum possible amount that can be awarded in this case is 1883.04 thousand dollars or  $ 1.883 Million. This value is less than the previously maximum value which is $3.5 million.