Question
The least absolute deviation line equation for the data in the table is m = 0.1x + 2.9 (estimated with technology).x y1 36 2.50.5 84 17.5 1.512 910 1311 4What is the sum of the absolute deviations?(1 point)Responses13.613.624.524.526.46726.4672727Skip to navigation
Answers
GPT-4o mini
To find the sum of the absolute deviations for the given data and the least absolute deviation line equation \( m = 0.1x + 2.9 \), we need to compute the predicted \( y \) values for each \( x \) from the table, calculate the absolute deviations, and then sum them up.
1. **Calculate the predicted \( y \) values using the equation \( m = 0.1x + 2.9 \)**:
- For \( x = 36 \):
\[
y = 0.1(36) + 2.9 = 3.6 + 2.9 = 6.5
\]
- For \( x = 84 \):
\[
y = 0.1(84) + 2.9 = 8.4 + 2.9 = 11.3
\]
- For \( x = 17.5 \):
\[
y = 0.1(17.5) + 2.9 = 1.75 + 2.9 = 4.65
\]
- For \( x = 1.5 \):
\[
y = 0.1(1.5) + 2.9 = 0.15 + 2.9 = 3.05
\]
- For \( x = 12 \):
\[
y = 0.1(12) + 2.9 = 1.2 + 2.9 = 4.1
\]
- For \( x = 910 \):
\[
y = 0.1(910) + 2.9 = 91 + 2.9 = 93.9
\]
- For \( x = 13 \):
\[
y = 0.1(13) + 2.9 = 1.3 + 2.9 = 4.2
\]
- For \( x = 4 \):
\[
y = 0.1(4) + 2.9 = 0.4 + 2.9 = 3.3
\]
2. **Calculate the absolute deviations:**
- The absolute deviation is given by \( |y_{\text{actual}} - y_{\text{predicted}}| \). Let's calculate these for each data point:
- For \( x = 36, y = 2.5 \):
\[
|2.5 - 6.5| = 4.0
\]
- For \( x = 84, y = 17.5 \):
\[
|17.5 - 11.3| = 6.2
\]
- For \( x = 17.5, y = 1.5 \):
\[
|1.5 - 4.65| = 3.15
\]
- For \( x = 1.5, y = 12 \):
\[
|12 - 3.05| = 8.95
\]
- For \( x = 12, y = 9 \):
\[
|9 - 4.1| = 4.9
\]
- For \( x = 910, y = 13 \):
\[
|13 - 93.9| = 80.9
\]
- For \( x = 4, y = 1 \):
\[
|1 - 3.3| = 2.3
\]
3. **Sum the absolute deviations**:
\[
\text{Sum of absolute deviations} = 4.0 + 6.2 + 3.15 + 8.95 + 4.9 + 80.9 + 2.3
\]
Let's calculate that:
\[
\text{Sum} = 4.0 + 6.2 = 10.2
\]
\[
10.2 + 3.15 = 13.35
\]
\[
13.35 + 8.95 = 22.3
\]
\[
22.3 + 4.9 = 27.2
\]
\[
27.2 + 80.9 = 108.1
\]
\[
108.1 + 2.3 = 110.4
\]
The total sum of the absolute deviations is \( 110.4 \).
Since the given options do not include 110.4, please double-check the provided lines and/or calculations. If further clarification is needed, do let me know!
1. **Calculate the predicted \( y \) values using the equation \( m = 0.1x + 2.9 \)**:
- For \( x = 36 \):
\[
y = 0.1(36) + 2.9 = 3.6 + 2.9 = 6.5
\]
- For \( x = 84 \):
\[
y = 0.1(84) + 2.9 = 8.4 + 2.9 = 11.3
\]
- For \( x = 17.5 \):
\[
y = 0.1(17.5) + 2.9 = 1.75 + 2.9 = 4.65
\]
- For \( x = 1.5 \):
\[
y = 0.1(1.5) + 2.9 = 0.15 + 2.9 = 3.05
\]
- For \( x = 12 \):
\[
y = 0.1(12) + 2.9 = 1.2 + 2.9 = 4.1
\]
- For \( x = 910 \):
\[
y = 0.1(910) + 2.9 = 91 + 2.9 = 93.9
\]
- For \( x = 13 \):
\[
y = 0.1(13) + 2.9 = 1.3 + 2.9 = 4.2
\]
- For \( x = 4 \):
\[
y = 0.1(4) + 2.9 = 0.4 + 2.9 = 3.3
\]
2. **Calculate the absolute deviations:**
- The absolute deviation is given by \( |y_{\text{actual}} - y_{\text{predicted}}| \). Let's calculate these for each data point:
- For \( x = 36, y = 2.5 \):
\[
|2.5 - 6.5| = 4.0
\]
- For \( x = 84, y = 17.5 \):
\[
|17.5 - 11.3| = 6.2
\]
- For \( x = 17.5, y = 1.5 \):
\[
|1.5 - 4.65| = 3.15
\]
- For \( x = 1.5, y = 12 \):
\[
|12 - 3.05| = 8.95
\]
- For \( x = 12, y = 9 \):
\[
|9 - 4.1| = 4.9
\]
- For \( x = 910, y = 13 \):
\[
|13 - 93.9| = 80.9
\]
- For \( x = 4, y = 1 \):
\[
|1 - 3.3| = 2.3
\]
3. **Sum the absolute deviations**:
\[
\text{Sum of absolute deviations} = 4.0 + 6.2 + 3.15 + 8.95 + 4.9 + 80.9 + 2.3
\]
Let's calculate that:
\[
\text{Sum} = 4.0 + 6.2 = 10.2
\]
\[
10.2 + 3.15 = 13.35
\]
\[
13.35 + 8.95 = 22.3
\]
\[
22.3 + 4.9 = 27.2
\]
\[
27.2 + 80.9 = 108.1
\]
\[
108.1 + 2.3 = 110.4
\]
The total sum of the absolute deviations is \( 110.4 \).
Since the given options do not include 110.4, please double-check the provided lines and/or calculations. If further clarification is needed, do let me know!