91Ó°ÊÓ

In 1997 a woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard (Genessy \(v\). Digital Equipment Corp.). 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 \(\$ 1000 \mathrm{~s}\) ) \(37,60,75,115,135,140,149,150,238,290\), \(340,410,600,750,750,750,1050,1100,1139,1150,1200\), \(1200,1250,1576,1700,1825\), and 2000 , from which \(\Sigma x_{i}=20,179, \Sigma x_{i}^{2}=24,657,511\). What is the maximum possible amount that could be awarded under the twostandard-deviation rule?

Step by step solution

01

Calculate the Mean

To find the mean of the awards, sum all the awards and divide by the number of cases. We are given \( \Sigma x_i = 20,179 \) and the number of cases is 27. The mean \( \mu \) is calculated as follows:\[ \mu = \frac{\Sigma x_i}{27} = \frac{20,179}{27} \approx 747.37 \]

02

Calculate the Standard Deviation

First, use the formula for variance which involves \( \Sigma x_i \) and \( \Sigma x_i^2 \). The variance \( \sigma^2 \) is given by:\[ \sigma^2 = \frac{\Sigma x_i^2}{n} - \left(\frac{\Sigma x_i}{n}\right)^2 \]\[ \sigma^2 = \frac{24,657,511}{27} - \left(\frac{20,179}{27}\right)^2 \]Calculating each term:\[ \frac{24,657,511}{27} \approx 912,500.41 \]\[ \left(\frac{20,179}{27}\right)^2 \approx 558,555.4 \]Then the variance:\[ \sigma^2 = 912,500.41 - 558,555.4 \approx 353,945.01 \]Now, calculate the standard deviation \( \sigma \):\[ \sigma = \sqrt{353,945.01} \approx 595.77 \]

03

Apply the Two Standard Deviation Rule

The maximum award allowed is two standard deviations above the mean. Therefore, we calculate:\[ \text{Maximum Award} = \mu + 2\sigma = 747.37 + 2(595.77) \]\[ \text{Maximum Award} = 747.37 + 1,191.54 = 1,938.91 \]

04

Round to the Nearest Thousands in Original Units

The calculated maximum award \( 1,938.91 \) is in thousands. Therefore, round it to the nearest thousand to determine the whole number amount of the award in the context of nearest possible thousands.Hence the award can be \( 1,939 \times 1,000 = 1,939,000 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation

The standard deviation is a measure of how spread out the numbers are in a data set. It gives us an idea of how much individual data points deviate from the mean or average. In the context of awards sums in the given exercise, understanding the standard deviation helps determine the range of amounts deemed reasonable by the court.

To calculate the standard deviation, you first need the variance of the data set. The variance represents the average squared difference from the mean. Once you have the variance, the standard deviation is simply its square root. This makes it much easier to apply and understand intuitively.

In this exercise, we see the variance calculated as follows:

• First, find the average of the squared sums divided by the number of data points.
• Subtract the square of the mean from this value to get the variance.
• The standard deviation is the square root of this variance.

In our case, the calculated standard deviation lets the court set limits on what constitutes a reasonable award by using multiples of this deviation.

Mean Calculation

The mean, often referred to as the average, is the central value of a set of numbers. It is a fundamental concept in statistics, providing a basic measure of the center of the data set. For the awards data in this problem, the mean represents the central point around which the court evaluates which compensations are deemed normative.

To find the mean, sum up all the awards and then divide this sum by the total number of awards. This provides a single value that represents the typical amount awarded in similar cases.

In our example:

• The sum of awards, provided as \( \Sigma x_i = 20,179 \), was divided by 27, reflecting the total number of cases.
• The resulting mean \( \mu \approx 747.37 \) is then used as the central benchmark for further calculations.

The mean serves as the foundation for calculating other statistics such as variance and standard deviation, hence its importance cannot be overstated in statistical analysis.

Variance

Variance is a statistical measure that describes the spread of data points within a data set. It measures how far each number in the set is from the mean and thus from every other number in the set. It is especially useful when describing the distribution of data as in our awards case.

The calculation of variance involves the following steps:

• First, find the square of the average values, computed as \( \left(\frac{\Sigma x_i}{n}\right)^2 \).
• Then, compute the average squared sums, shown as \( \frac{\Sigma x_i^2}{n} \).
• The variance \( \sigma^2 \) is the difference between these two values.

This measure of variance (\( \sigma^2 \approx 353,945.01 \)) demonstrates how much the awards deviate from the average, laying the groundwork for determining the extent of reasonable awards in this situation. Understanding variance is crucial to grasping the bigger picture of statistical significance in any distribution.