Question

A small town has an adult population of \(100 .\) One person is a multimillionaire who makes \(\$ 5,000,000\) per year. The mean annual salary in the town is \(\$ 74,750 .\) Is the mean a fair measure of a typical salary in the town? What is the mean annual salary of the other 99 adults?

Short answer

The mean salary of the other 99 adults in the town is approximately \$25,000. The mean is not a fair measure of a typical salary in the town as it's heavily influenced by the multimillionaire's income.

Step-by-step solution

Calculating the Total Income of the Town

The mean annual salary of the town, which includes 100 adults, is given as $74,750. To find the total income of the town we multiply the mean salary with the total number of adults. Thus, the total income of the town equals \(100 \times \$74,750 = \$7,475,000\).

Calculate the Combined Salary of the 99 Adults

The total salary of the town already includes the salary of the multimillionaire. To find the total salary of the rest of the 99 adults, we need to subtract the multimillionaire's income from the total income of the town. So, the combined salary of the other 99 adults equals \(\$7,475,000 - \$5,000,000 = \$2,475,000\).

Calculate the Mean Salary of the 99 Adults

The mean salary is calculated by dividing the total income by the number of individuals. Thus, the mean annual salary of the other 99 adults equals \(\$2,475,000 / 99 = \$25,000\ approximately.

Analyze the Mean as a Measure of Typical Salary

In this case, the mean salary of the town is strongly influenced by the income of the multimillionaire, which skews the mean towards a much higher value. Hence, it can be concluded that the mean is not a fair measure of a typical salary in the town, since the income of the multimillionaire significantly skews the average, not giving a true reflection of the other residents' income.