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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 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 two standard deviations of the mean of the awards in the 27 cases. The 27 awards were (in \(\$ 1000\) 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 \(\sum x_{i}=20,179, \sum x_{i}^{2}=\) \(24,657,511\). What is the maximum possible amount that could be awarded under the twostandard- deviation rule?

Step by step solution

01

Understand the Problem

To find the maximum possible award within two standard deviations of the mean, we need to calculate the mean and standard deviation of the given awards and then determine the range allowed by the two-standard-deviation rule. This rule states that a reasonable award is within two standard deviations above or below the mean.

02

Calculate the Mean

The mean (\(\mu\)) is calculated by dividing the sum of all awards (\(\sum x_i\)) by the number of awards (N). Here, \(\sum x_i = 20179\) and \(N = 27\).\[\mu = \frac{\sum x_i}{N} = \frac{20179}{27}\]Calculate \(\mu\)\.

03

Calculate the Mean Step Result

Compute the division:\[\mu = \frac{20179}{27} \approx 747.37\]so, the mean award is approximately \(747.37\) thousands of dollars.

04

Calculate the Variance

The variance \(\sigma^2\) is given by \(\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2\). Use \(\sum x_i^2 = 24657511\) and \(N = 27\)\.\[\sigma^2 = \frac{24657511}{27} - (747.37)^2\]Calculate \(\sigma^2\)\.

05

Calculate the Variance Step Result

First compute: \[\frac{24657511}{27} \approx 912500.41\]Then compute \(\sigma^2\) by subtracting the square of the mean:\[\sigma^2 = 912500.41 - (747.37)^2\]\(\approx 912500.41 - 558561.27 = 353939.14\)Thus, the variance is approximately \(353939.14\).

06

Calculate the Standard Deviation

Calculate the standard deviation \(\sigma\) by taking the square root of the variance:\[\sigma = \sqrt{353939.14}\]Calculate \(\sigma\)\.

07

Calculate the Standard Deviation Step Result

Compute the square root:\[\sigma \approx \sqrt{353939.14} \approx 595.67\]Thus, the standard deviation is approximately \(595.67\).Use this to establish the two-standard-deviation range.

08

Determine the Maximum Award Within Two Standard Deviations

The maximum award within two standard deviations above the mean is given by:\(\mu + 2\sigma = 747.37 + 2 \cdot 595.67\)Compute this value.

09

Final Calculation for Maximum Award

Calculate:\[747.37 + 2 \times 595.67 = 747.37 + 1191.34 = 1938.71\]Thus, the maximum possible award under the two-standard-deviation rule is about \(1938.71\) thousand dollars.

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Key Concepts

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

Mean Calculation

The mean, often referred to as the average, is a fundamental concept in statistical analysis. It provides a measure of central tendency, which essentially is a single value representing the center of a dataset. To calculate the mean, you sum up all the data values and divide by the number of data points. In this exercise, the sum of the awards is given as \( \sum x_i = 20179 \), and there are 27 awards in total.

Thus, the mean \( \mu \) can be calculated using the formula:

• \( \mu = \frac{\sum x_i}{N} = \frac{20179}{27} \)

By performing the division, we find that the mean award is approximately 747.37 (in thousands of dollars). This value is crucial because it helps establish the center around which we will measure variations, ultimately leading to more complex analyses such as determining the reasonable range of data points.

Standard Deviation

The standard deviation is a key statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that data points are close to the mean, whereas a high standard deviation indicates they are spread out over a wider range.

To calculate the standard deviation \( \sigma \), you must first find the variance. The variance \( \sigma^2 \) is given by:

• \( \sigma^2 = \frac{\sum x_i^2}{N} - \mu^2 \)

Using the provided values \( \sum x_i^2 = 24657511 \) and the mean \( \mu = 747.37 \), compute:

• \( \sigma^2 = \frac{24657511}{27} - (747.37)^2 \approx 353939.14 \)

The square root of the variance gives the standard deviation:

• \( \sigma = \sqrt{353939.14} \approx 595.67 \)

This standard deviation tells us that on average, the awards vary by about 595.67 thousand dollars from the mean.

Variance

Variance is another crucial concept in statistical analysis that helps us understand the degree of spread in the dataset. It essentially measures how far each number in the set is from the mean, thus giving us insight into the data's variability.

The variance \( \sigma^2 \) is calculated as:

• \( \sigma^2 = \frac{\sum x_i^2}{N} - \mu^2 \)

In our specific exercise, after calculating we found the variance to be approximately 353939.14. This number is vital because it forms the basis for calculating the standard deviation. By squaring the difference between each data point and the mean, we observe how much the data deviates on average. The higher the variance, the more spread out the data. This value gives us a comprehensive view of how awards vary among the cases considered.

Two-standard-deviation Rule

The two-standard-deviation rule is a statistical guideline used to determine the range within which a certain percentage of data points are expected to fall. It effectively defines a threshold for what is considered typical or normal within a data set. For a dataset following a normal distribution, approximately 95% of the data points lie within two standard deviations of the mean.

In this example, the maximum potential award was determined by adding two standard deviations to the mean:

• \( \mu + 2\sigma = 747.37 + 2 \times 595.67 \)
• \( = 1938.71 \)

Thus, the maximum award that could be considered reasonable, given the constraint, is about 1938.71 thousand dollars. This rule is crucial in practical scenarios where claims or outcomes should fall within an acceptable range, thereby balancing fairness and predictability with statistical rigor.