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Problem 48 Hard Difficulty

Washington Post writer John Schwartz pointed out that if Microsoft Corp. cofounder Bill Gates, who, at the time, was reportedly worth $\$ 10$ billion, lived in a town with 10,000 totally penniless people, the average personal wealth in the town would make it seem as if everyone were a millionaire. Source: The Washington Post.
a. Verify Schwartz’s statement.
b. What would be the median personal wealth in this town?
c. What would be the mode for the personal wealth in this town?
d. In this example, which average is most representative: the mean, the median, or the mode?

Answer

a. $\quad \overline{x}=\frac{10,000(0)+1(10,000,000,000)}{10,001}=999,900.01$ The mean wealth is approximately $\$ 1$ million.
b. 0 is the middle value in the list of $10,001$ (odd number of ) values. The median is $\$ 0$
c. The mode is $\$ 0,$ since it occurs with the greatest frequency.
d. The median or mode is most representative of the wealth of the town's population, since almost all have zero personal wealth.

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Video Transcript

this question gives a hypothetical scenario where Bill Gates, valued at $10 billion lives in a town with 10,000 destitute people with no money. Good. It's this promise set up like a frequency table. We have a couple of values. Some people have $0 others have 10 billion and were given their frequencies. We know that there are 10,000 people with $0 only one person with $10 billion. Finding the mean of this town now is like finding the meaning for frequency. Table explore is equal to the sum of X times f over to some of F. Now let's add a column for X times after this table. We know that zero times 10,000 is gonna be zero in one times 10 billion is just 10 billion. So are some of X times f is clearly just gonna be 10 billion. And this some of our frequencies is gonna be 10,000 and one so export 10 billion divided by 10,000 and one. If we plug this into our calculator, we'll get that thea average is now, um, 999,000. Ah, and 900 dollars. So are they? Billionaires are the millionaires? No. Are they billionaires? Absolutely. Are they almost millionaires? Absolutely. There. Almost. Millionaires. 999,000 is awfully close. So this would be that mean of the values in this town. But what would be the median? Well, if there are 10,000 zeros and 1 10 billion right in the middle is still gonna be zero, right? If we have 10,000 and one total data points, our, uh I mean will come from our median will be the 5/1000 the 5000 and first data point. So our median, the 5001st state of point, is still gonna be zero. That's her medium. Likewise, our mode, our mode, when there are 10,000 zeros and only 1 10 billion are mode is still zero. So we can see that are mean is very sensitive to extreme data. But the median and mode are not Is your final