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Jeffrey is serving on a six-person jury for a personal-injury lawsuit. All six jurors want to award damages to the plaintiff but cannot agree on the amount of the award. The jurors have decided that each of them will suggest an amount that he or she thinks should be awarded; then they will use the mean of these six numbers as the award to recommend to the plaintiff. a. Jeffrey thinks the plaintiff should receive \(\$ 20,000\), but he thinks the mean of the other five jurors' recommendations will be about \(\$ 12,000 .\) He decides to suggest an inflated amount so that the mean for all six jurors is \(\$ 20,000\). What amount would Jeffrey have to suggest? b. How might this jury revise its procedure to prevent a juror like Jeffrey from having an undue influence on the amount of damages to be awarded to the plaintiff?

Step by step solution

01

Calculate the total desired sum

Firstly, it's important to determine the total desired amount that all jurors combined should suggest so that the mean for all six jurors is $20,000. The mean is calculated by summing all the values and dividing by the number of values. In this case, the desired mean is already provided, so to get the total desired sum, multiply this mean by the number of jurors. Thus, it is \(20,000 * 6 = 120,000\)

02

Calculate the total sum of the other five jurors

To get the total expected sum of the recommendations of the other five jurors, multiply Jeffrey's expectation of their mean recommendation by their number. Thus, the result is \(12,000 * 5 = 60,000\)

03

Calculate Jeffrey’s inflated suggestion

To inflate his recommendation in such a way that the mean of all the six suggestions becomes $20,000, Jeffrey has to suggest an amount that is equal to the difference between the total desired sum and the sum of what the other five jurors are expected to suggest. This value can be found by subtracting the result of step 2 from the result of step 1. So \(120,000 - 60,000 = 60,000\). Jeffrey thus needs to suggest $60,000 to have the mean of all six jurors at $20,000.

04

Altering the procedure

To tackle undue influence from any juror, the jury could adopt the median average of the suggested amounts, rather than a simple arithmetic mean. The median is the number that separates the higher half and lower half of a data sample, so it will prevent extreme values from unduly affecting the result.

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Key Concepts

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

Mean Calculation

When solving statistical problems, one of the most common concepts is calculating the mean, or average. To find the mean, you sum up all the individual values and then divide the total by the number of values. This calculates the average, providing a single number that represents a set of data. For instance, if you want the mean award to be $20,000 among six jurors, you would multiply 20,000 by 6, resulting in 120,000. Therefore, the total sum of all suggested awards needs to equal 120,000 to achieve the desired mean.

Inflated Suggestion

An inflated suggestion refers to the act of providing a higher-than-normal value to skew the average. In this scenario, Jeffrey wishes to influence the jury's decision by suggesting a higher amount than he actually believes is fair. By doing this, he aims to manipulate the overall mean to match his desired outcome. To ensure the mean reaches $20,000, Jeffrey calculates the difference needed after considering the expected total from the other jurors. Here, he suggests $60,000, which is determined by subtracting the expected $60,000 from the other jurors from the desired $120,000 total.

Procedure Alteration

To prevent an undue influence like Jeffrey's, the selection procedure for deciding on the award amount needs adjustment. One effective method is to use a different approach instead of a simple mean. Switching from an arithmetic mean to a different calculation, such as using the median instead, is a great solution. The median is suitable because it finds the middle value in the list of numbers, which means that extremely high or low values do not heavily impact the final decision. This ensures that a single juror cannot disproportionately affect the decision, resulting in a more fair and representative outcome.

Median Average

The median average is a technique often used to find a value that represents a data set without being influenced by outliers. Unlike the mean, which sums up all values, the median focuses on the central value in a sorted list. Regardless of any extreme values suggested by the jurors, the median keeps the decision-making grounded in the majority's perspective. For the six jurors, finding the median means listing their suggestions in order and selecting the middle number. If the list has an even number of data points, the median is the average of the two middle numbers. This method encourages balanced and fair decision-making, safeguarding against one outlier influencing the jury's recommendation.