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In 1997, a woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard (Genessey v. Digital Equipment Corporation). The jury 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 2 standard deviations of the mean of the awards in the 27 cases. The 27 award amounts were (in thousands of dollars) \(\begin{array}{rrrrrrrr}37 & 60 & 75 & 115 & 135 & 140 & 149 & 150 \\ 238 & 290 & 340 & 410 & 600 & 750 & 750 & 750 \\\ 1050 & 1100 & 1139 & 1150 & 1200 & 1200 & 1250 & 1576 \\ 1700 & 1825 & 2000 & & & & & \end{array}\) What is the maximum possible amount that could be awarded under the "2-standard deviations rule?"

Step by step solution

01

Calculate the Mean

We first need to calculate the mean (average) of the awards. The formula to calculate the mean is \[Mean = \frac{Sum\: of\: all\: awards}{Number\: of\: awards}\]. Add all the given awards together and then divide by the total number of awards which in this case is 27.

02

Calculate the Standard Deviation

The standard deviation is a measure of dispersion in a set of data. For calculating the standard deviation, we first find the difference of each award from the mean, square these differences, add them up, divide the total by the number of observations (n-1, where n is the total number of awards), and then take the square root of this quotient. The formula is given as \[Standard\: Deviation = \sqrt{\frac{\Sigma(X - Mean)^2}{n-1}}\]. This will give us the dispersion of the awards.

03

Calculate the Maximum Possible Award

Once the mean and standard deviation have been calculated, we then calculate the maximum possible award under the '2 standard deviations' rule. This is simply the mean plus twice the standard deviation. Hence, \[Max\: award = Mean + 2 * Standard\: Deviation\]. This value will give us the maximum reasonable award under the rule specified by the court.

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Key Concepts

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

Mean Calculation

When trying to find the mean, or average, of a set of numbers, you're essentially figuring out what the central or typical value is across all data points. It's like trying to find the middle ground that represents the overall pattern of the data.

Here's how you do it:

• Add all the numbers together. This gives you a total sum.
• Next, count how many numbers there are.
• Divide the total sum by the number of data points.

For example, if you're calculating the mean award from the 27 cases in the lawsuit, you'd add all the awards together. Then, you'd divide that total by 27, since there are 27 awards in total.

This gives you the mean, serving as a simple measure to indicate the average amount awarded.

Dispersion of Data

Dispersion in data tells us how spread out the values in a dataset are. Imagine throwing a handful of coins onto a table. If they all land in one small area, they're closely packed, but if they land all over the place, they're more dispersed.

Standard deviation is one way to measure this dispersion. It tells us how much the values differ from the mean (average).

To calculate standard deviation:

• Find the difference between each data point and the mean.
• Square each of these differences.
• Add up all these squared differences.
• Divide this sum by the number of data points minus one (n-1).
• Finally, take the square root of this quotient.

The higher the standard deviation, the more spread out the data is around the mean. Conversely, a smaller standard deviation means the data points are closer to the mean. This helps determine how consistent or variable the awards are across the different cases.

Two Standard Deviations Rule

The 'Two Standard Deviations Rule' is a handy guideline used to establish a range within which the data usually falls. This rule is especially useful in court determinations, like in the lawsuit mentioned.

Here's how it works:

• Start with the mean.
• Then, calculate the standard deviation.
• Multiply the standard deviation by two.
• Add this result to the mean to determine the maximum reasonable value.

So, if a court decides that an award is only reasonable if it's within two standard deviations of the mean, this defines a benchmark for deciding if an award is unusually high or low.

In practical terms, this means about 95% of your data should lie within this range. It helps ensure that decisions are based on a reasonable, typical range derived from the data, thereby reducing the likelihood of extreme outliers influencing the outcome.