Question

The least absolute deviation line equation for the data in the table is m = 0.15x + 3 (estimated with technology).x y1 36 2.50.5 84 17.5 1.512 910 1311 4What is the sum of the absolute deviations?

Answer 1

To find the sum of the absolute deviations for the least absolute deviation line represented by the equation \( m = 0.15x + 3 \), we need to calculate the predicted values of \( y \) based on the given \( x \) values and the line equation, and then compute the absolute differences between the actual \( y \) values and the predicted \( y \) values.

The given data points are:
1. \( (x_1, y_1) = (36, 2.5) \)
2. \( (x_2, y_2) = (0.5, 84) \)
3. \( (x_3, y_3) = (17.5, 1.5) \)
4. \( (x_4, y_4) = (12, 910) \)
5. \( (x_5, y_5) = (13, 11) \)

Now, we will substitute each \( x \) value into the equation \( m = 0.15x + 3 \) to find the predicted \( y \) values and then calculate the absolute deviations.

1. **For \( x = 36 \)**:
\[
m = 0.15(36) + 3 = 5.4 + 3 = 8.4
\]
Absolute deviation: \( |2.5 - 8.4| = 5.9 \)

2. **For \( x = 0.5 \)**:
\[
m = 0.15(0.5) + 3 = 0.075 + 3 = 3.075
\]
Absolute deviation: \( |84 - 3.075| = 80.925 \)

3. **For \( x = 17.5 \)**:
\[
m = 0.15(17.5) + 3 = 2.625 + 3 = 5.625
\]
Absolute deviation: \( |1.5 - 5.625| = 4.125 \)

4. **For \( x = 12 \)**:
\[
m = 0.15(12) + 3 = 1.8 + 3 = 4.8
\]
Absolute deviation: \( |910 - 4.8| = 905.2 \)

5. **For \( x = 13 \)**:
\[
m = 0.15(13) + 3 = 1.95 + 3 = 4.95
\]
Absolute deviation: \( |11 - 4.95| = 6.05 \)

Now, we sum the absolute deviations:
\[
5.9 + 80.925 + 4.125 + 905.2 + 6.05 = 1002.2
\]

Thus, the sum of the absolute deviations is \( \boxed{1002.2} \).