Question

For Exercises 4-7, refer to the table that shows the recent population,in millions, of the ten largest U.S. cities.4. Find the mean absolute deviation. Round to the nearest hundredth.Population of LargestU.S. Cities (millions)1.5/3.8 1.3, 1.6 2.91.4 0.9 2.38.41.3,15 How many data values are closer than one mean absolute deviationaway from the mean?6. Which population is farthest from the mean? How far away from the meanis that population? Round to the nearest hundredth.7. Are there any populations that are more than twice the mean absolutedeviation from the mean? Explain.

Answer 1

• 4). The mean absolute deviation is 1.50 million (rounded to the nearest hundredth).

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• 5). 8 data values

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• 6). 8.4 million, 5.86 million

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• 7)Yes, 8.4 million

Explanation:

Part 4

Given the data, first, we find the mean.

`\begin{gathered} Sum=1.5+3.8+1.3+1.6+2.9+ \\ 1.4+0.9+2.3+8.4+1.3=25.4 \\ \implies Mean=(25.4)/(10)=2.54 \end{gathered}`

Next, subtract the mean from each of the data, take the absolute value and sum:

`\begin{gathered} Absolute\;Sum=|1.5-2.54|+|3.8-2.54|+|1.3-2.54|+|1.6-2.54|+|2.9-2.54| \\ +|1.4-2.54|+|0.9-2.54|+|2.3-2.54|+|8.4-2.54|+|1.3-2.54| \\ =1.04+1.26+1.24+0.94+0.36+1.14+1.64+0.24+5.86+1.24 \\ =14.96 \end{gathered}`

Therefore, the mean absolute deviation is:

`M.A.D=(14.96)/(10)=1.50\text{ \lparen rounded to the nearest hundredth\rparen}`

The mean absolute deviation is 1.50 million (rounded to the nearest hundredth).

Part 5

• The mean = 2.54 million

,

• The mean absolute deviation = 1.50 million

There are 8 data values that are closer than one mean absolute deviation away from the mean.

Part 6

The population that is farthest from the mean = 8.4 million

The distance away from the mean = |8.4-2.54| = 5.86 million.

Part 7

Twice the mean absolute deviation = 2 x 1.50 million = 3.00 million

Since 5.86 million > 3.00 million, the population of 8.4 million is greater than 3.00 million away from the mean.

Thus, the population 8.4 million is more than twice the mean absolute deviation from the mean.