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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

The mean of a data set is the sum of all data values divided by the number of data values. Let's denote the given data values as \(x_i\), for \(i = 1, 2, ..., 27\). Then the mean is calculated as \[ \text{mean} = \frac{\sum_{i=1}^{27}x_i}{27} \]

02

Calculate the standard deviation

The standard deviation of a data set is calculated using the following formula: \[ \text{standard deviation} = \sqrt{\frac{\sum_{i=1}^{27}(x_i - \text{mean})^2}{27-1}} \] where \(x_i - \text{mean}\) is the deviation of each data value from the mean.

03

Apply the 2-standard deviation rule

According to the court's rule, the award should be within two standard deviations of the mean. Therefore, the maximum possible award is calculated by adding two times the standard deviation to the mean. \[ \text{maximum possible award} = \text{mean} + 2 \times \text{standard deviation} \] This value provides the maximum possible amount that could be awarded in such a case according to the specified rule.

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Key Concepts

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

Statistical Analysis

Statistical analysis plays a crucial role in interpreting data, making decisions, and solving real-world problems. In the court case mentioned, statistical analysis was used to determine reasonsable compensation for pain and suffering by looking at past cases. It involves collecting, summarizing, and interpreting data to extract meaningful insights. Descriptive statistics, which include measures like the mean and standard deviation, are used to summarize the main features of a dataset. In more complex analyses, statisticians may use inferential statistics to draw conclusions about a larger population from a sample. In this case, the mean and standard deviation are vital in establishing a benchmark for reasonable award amounts, ensuring decisions aren't based on arbitrary judgements but on a clear analytical method.

Normal Distribution

The concept of normal distribution, often referred to as the bell curve, is pivotal in statistics. Many real-world phenomena tend to distribute in a pattern that mirrors the normal distribution, which is symmetric about the mean and characterized by its bell-shaped curve. This distribution allows statisticians to make probabilistic statements about data. For instance, in the legal case scenario, assuming that the award amounts are normally distributed, using the '2-standard deviations rule' helps determine that approximately 95% of the cases will fall within that range from the mean. This aids the court in defining what a 'reasonable' award might look like, as they can infer that awards outside of this range are exceptionally high or low.

Descriptive Statistics

Descriptive statistics consist of key measures that describe a dataset's central tendency, dispersion, and shape. The mean, median, mode, range, variance, and standard deviation are all fundamental tools in this branch of statistics. In the context of the lawsuit, the court used these statistics to analyze compensation across similar cases. The mean provides a central value, while the standard deviation measures how much variation or dispersion exists from the mean. These two measures alone can offer valuable insights into what is typical or atypical in the context of legal compensation amounts.

Mean Calculation

Mean calculation is a basic yet mighty element of descriptive statistics, representing the average value in a dataset. It's calculated by summing up all the values and dividing by the total number of values. In the legal case scenario presented, calculating the mean of the award amounts gives insight into what is standard across the 27 cases considered relevant. Understanding how to calculate the mean is essential, as it is a benchmark for determining reasonable compensation using the '2-standard deviations rule'. The simplicity of mean calculation is deceptive; it is a powerful tool to create a baseline for comparison and further statistical analysis.