Answer:
<h2>A. The difference of the means is 1. This value is less than half of the mean absolute deviation of either data set.</h2>
Step-by-step explanation:
The mean absolute deviation is defined by
![MAD=\frac{\sum (x- \mu)}{N}](https://tex.z-dn.net/?f=MAD%3D%5Cfrac%7B%5Csum%20%28x-%20%5Cmu%29%7D%7BN%7D)
Where
is the mean and
is the total number of elements.
First, we find each mean.
<h3>Blue data set.</h3>
![\mu_{blue} = \frac{1+2+4+4+5+5+6+7+9+9+9+11}{12} =\frac{72}{12}= 6](https://tex.z-dn.net/?f=%5Cmu_%7Bblue%7D%20%20%3D%20%5Cfrac%7B1%2B2%2B4%2B4%2B5%2B5%2B6%2B7%2B9%2B9%2B9%2B11%7D%7B12%7D%20%3D%5Cfrac%7B72%7D%7B12%7D%3D%206)
<h3>Green data set.</h3>
![\mu_{green}=\frac{1+3+4+6+6+6+7+9+9+10+10+13}{12}= \frac{84}{12}=7](https://tex.z-dn.net/?f=%5Cmu_%7Bgreen%7D%3D%5Cfrac%7B1%2B3%2B4%2B6%2B6%2B6%2B7%2B9%2B9%2B10%2B10%2B13%7D%7B12%7D%3D%20%20%5Cfrac%7B84%7D%7B12%7D%3D7)
As you can observe, they have a difference of 1 unit regarding their means.
Now, let's find each MAD.
<h3>Blue data set.</h3>
First, we find the difference between the mean and each data, to then sum all differences.
1 - 6 = |-5|
2 - 6 = |-4|
4 - 6 =|-2|
4 - 6 = |-2|
5 - 6 = |-1|
5 - 6 =|-1|
6 - 6 = 0
7 - 6 = |1|
9 - 6 = |3|
9 - 6 = |3|
9 - 6 =| 3|
11 - 6 =| 5|
Which gives a total of 30.
Then,
![MAD=\frac{30}{12}=2.5](https://tex.z-dn.net/?f=MAD%3D%5Cfrac%7B30%7D%7B12%7D%3D2.5)
<h3>Green data set.</h3>
We repeat the process.
1 - 7 = |-6|
3 - 7 =|-4|
4 - 7 = |-3|
6 - 7 = |-1|
6 - 7 = |-1|
6 - 7 =| -1|
7 - 7 = 0
9 - 7 = |2|
9 - 7 = |2|
10 - 7 =| 3|
10 - 7 = |3|
13 - 7 =|6|
Which gives a total of 32.
Then,
![MAD=\frac{32}{12} \approx 2.67](https://tex.z-dn.net/?f=MAD%3D%5Cfrac%7B32%7D%7B12%7D%20%5Capprox%202.67)
Notice that half of each mean is greater than one.
Therefore, the choice A is correct.