The least absolute deviation line equation for the data in the table is m = 0.1 x + 2.9

X 1,6,0.5,4,7.5,12,10,11 y 3,2.5,8,1,1.5,9,13,4

What is the sum of the absolute deviations?

First find the absolute deviation for each coordinate.

(1,3), (6,2.5), (0.5,8), (4,1), (7.5,1.5), (12,9), (10,13), (11,4).
I will do the 1st coordinate: (1,3)

Least absolute deviation line equation: m(x)= 0.1 x + 2.9
If you plot all 8 coordinates onto a xy graph, m(x) is the line of best fit.
The distance from one coordinate's y-value from the line m(x) is the deviation.

Plug in x from coordinate (1,3) into the equation m(1):
At x = 1, m(1)= 0.1(1) + 2.9 = 3.0
This is (1,3) on the m(x) line.

Find the distance of coordinate (1,3) from (1,3):
Absolute Deviation = |m(1) - y| = |3 - 3| = 0

Repeat for 2nd coordinate: (6, 2.5)
m(6) = 0.1(6) + 2.9 = 3.5
(6,3.5) on the m(x) line.
Absolute Deviation: |m(6) - y| = |3.5 - 2.5| = 1.0

Repeat this again for the other six coordinates.
Add up the eight absolute deviations to find the sum.

Well, if I could borrow a calculator from you, I'd be able to compute the sum of the absolute deviations. The only thing I can sum up right now is my clown shoes and my comedic timing!

To find the sum of the absolute deviations, we need to calculate the absolute deviation for each data point and then sum them up.

The absolute deviation of a data point from the least absolute deviation line is the absolute value of the difference between the observed y-value (from the table) and the predicted y-value (calculated using the equation m = 0.1x + 2.9).

Using the given equation,

m = 0.1x + 2.9

Calculating the predicted y-values for each x-value:

For x = 1, y = 0.1(1) + 2.9 = 0.1 + 2.9 = 3

For x = 6, y = 0.1(6) + 2.9 = 0.6 + 2.9 = 3.5

For x = 0.5, y = 0.1(0.5) + 2.9 = 0.05 + 2.9 = 2.95

For x = 4, y = 0.1(4) + 2.9 = 0.4 + 2.9 = 3.3

For x = 7.5, y = 0.1(7.5) + 2.9 = 0.75 + 2.9 = 3.65

For x = 12, y = 0.1(12) + 2.9 = 1.2 + 2.9 = 4.1

For x = 10, y = 0.1(10) + 2.9 = 1 + 2.9 = 3.9

For x = 11, y = 0.1(11) + 2.9 = 1.1 + 2.9 = 4

Now, calculate the absolute deviation for each data point by subtracting the predicted y-value from the observed y-value and taking the absolute value:

|3 - 3| = 0
|2.5 - 3.5| = 1
|8 - 2.95| = 5.05
|1 - 3.3| = 2.3
|1.5 - 3.65| = 2.15
|9 - 4.1| = 4.9
|13 - 3.9| = 9.1
|4 - 4| = 0

Now, sum up the absolute deviations:

0 + 1 + 5.05 + 2.3 + 2.15 + 4.9 + 9.1 + 0 = 24.5

Therefore, the sum of the absolute deviations is 24.5.

To find the sum of the absolute deviations, we need to calculate the absolute deviation for each data point, and then sum up all the individual absolute deviations.

The absolute deviation of a data point is the absolute value of the difference between the actual value and the predicted value. In this case, the "predicted value" is given by the equation m = 0.1x + 2.9.

Let's calculate the absolute deviation for each data point:

For x = 1: the predicted value is m = 0.1(1) + 2.9 = 3
The absolute deviation is |3 - 3| = 0

For x = 6: the predicted value is m = 0.1(6) + 2.9 = 3.5
The absolute deviation is |2.5 - 3.5| = 1

For x = 0.5: the predicted value is m = 0.1(0.5) + 2.9 = 3.04
The absolute deviation is |8 - 3.04| = 4.96

For x = 4: the predicted value is m = 0.1(4) + 2.9 = 3.3
The absolute deviation is |1 - 3.3| = 2.3

For x = 7.5: the predicted value is m = 0.1(7.5) + 2.9 = 3.65
The absolute deviation is |1.5 - 3.65| = 2.15

For x = 12: the predicted value is m = 0.1(12) + 2.9 = 4.1
The absolute deviation is |9 - 4.1| = 4.9

For x = 10: the predicted value is m = 0.1(10) + 2.9 = 4.9
The absolute deviation is |13 - 4.9| = 8.1

For x = 11: the predicted value is m = 0.1(11) + 2.9 = 5
The absolute deviation is |4 - 5| = 1

Now, we add up all the absolute deviations:

0 + 1 + 4.96 + 2.3 + 2.15 + 4.9 + 8.1 + 1 = 24.41

Therefore, the sum of the absolute deviations for this given data is 24.41.