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)Responses 13.613.624.524.526.46726.46727
Answers
GPT-4o mini
To find the sum of the absolute deviations for the least absolute deviation (LAD) line equation \( m = 0.1x + 2.9 \) given the data points, we need to follow these steps:
1. Calculate the predicted \( y \) values using the equation for each corresponding \( x \).
2. Compute the absolute deviations, which is the absolute difference between the actual \( y \) values and the predicted \( y \) values.
3. Sum all the absolute deviations.
Let's calculate this step by step using the provided data points.
### Given Data Points
| \( x \) | \( y \) |
|----------|----------|
| 36 | 2.5 |
| 0.5 | 84 |
| 17.5 | 1.5 |
| 12 | 9 |
| 10 | 13 |
| 11 | 4 |
### Step 1: Calculate Predicted \( y \) Values
Using the equation \( y = 0.1x + 2.9 \):
- For \( x = 36 \):
\[ y = 0.1(36) + 2.9 = 3.6 + 2.9 = 6.5 \]
- For \( x = 0.5 \):
\[ y = 0.1(0.5) + 2.9 = 0.05 + 2.9 = 2.95 \]
- For \( x = 17.5 \):
\[ y = 0.1(17.5) + 2.9 = 1.75 + 2.9 = 4.65 \]
- For \( x = 12 \):
\[ y = 0.1(12) + 2.9 = 1.2 + 2.9 = 4.1 \]
- For \( x = 10 \):
\[ y = 0.1(10) + 2.9 = 1.0 + 2.9 = 3.9 \]
- For \( x = 11 \):
\[ y = 0.1(11) + 2.9 = 1.1 + 2.9 = 4.0 \]
### Step 2: Compute Absolute Deviations
Now, let’s calculate the absolute deviations from the actual \( y \) values:
| \( x \) | Actual \( y \) | Predicted \( y \) | Absolute Deviation |
|----------|----------------|--------------------|---------------------|
| 36 | 2.5 | 6.5 | \( |2.5 - 6.5| = 4.0 \) |
| 0.5 | 84 | 2.95 | \( |84 - 2.95| = 81.05 \) |
| 17.5 | 1.5 | 4.65 | \( |1.5 - 4.65| = 3.15 \) |
| 12 | 9 | 4.1 | \( |9 - 4.1| = 4.9 \) |
| 10 | 13 | 3.9 | \( |13 - 3.9| = 9.1 \) |
| 11 | 4 | 4.0 | \( |4 - 4.0| = 0.0 \) |
### Step 3: Sum of Absolute Deviations
Now, we sum up the absolute deviations:
\[
4.0 + 81.05 + 3.15 + 4.9 + 9.1 + 0.0 = 102.2
\]
This means the sum of the absolute deviations is \( 102.2 \).
However, this result seems inconsistent with the options provided in the multiple-choice format. The process I've performed gives the sum of absolute deviations accurately calculated based on the formula.
There may be more context needed, or there could have been a mistake in the given values from the task statement. Nevertheless, per calculations, the sum of absolute deviations would equal \( 102.2 \).
If you're looking for a multiple-choice option, none of the provided options (13.6, 13.62, 24.5, 24.52, 26.46, 72, 26.467) matches the calculated sum of \( 102.2 \). I recommend reviewing to ensure that the data or line equation is accurate, or consider whether or not actual values were correctly transcribed.
1. Calculate the predicted \( y \) values using the equation for each corresponding \( x \).
2. Compute the absolute deviations, which is the absolute difference between the actual \( y \) values and the predicted \( y \) values.
3. Sum all the absolute deviations.
Let's calculate this step by step using the provided data points.
### Given Data Points
| \( x \) | \( y \) |
|----------|----------|
| 36 | 2.5 |
| 0.5 | 84 |
| 17.5 | 1.5 |
| 12 | 9 |
| 10 | 13 |
| 11 | 4 |
### Step 1: Calculate Predicted \( y \) Values
Using the equation \( y = 0.1x + 2.9 \):
- For \( x = 36 \):
\[ y = 0.1(36) + 2.9 = 3.6 + 2.9 = 6.5 \]
- For \( x = 0.5 \):
\[ y = 0.1(0.5) + 2.9 = 0.05 + 2.9 = 2.95 \]
- For \( x = 17.5 \):
\[ y = 0.1(17.5) + 2.9 = 1.75 + 2.9 = 4.65 \]
- For \( x = 12 \):
\[ y = 0.1(12) + 2.9 = 1.2 + 2.9 = 4.1 \]
- For \( x = 10 \):
\[ y = 0.1(10) + 2.9 = 1.0 + 2.9 = 3.9 \]
- For \( x = 11 \):
\[ y = 0.1(11) + 2.9 = 1.1 + 2.9 = 4.0 \]
### Step 2: Compute Absolute Deviations
Now, let’s calculate the absolute deviations from the actual \( y \) values:
| \( x \) | Actual \( y \) | Predicted \( y \) | Absolute Deviation |
|----------|----------------|--------------------|---------------------|
| 36 | 2.5 | 6.5 | \( |2.5 - 6.5| = 4.0 \) |
| 0.5 | 84 | 2.95 | \( |84 - 2.95| = 81.05 \) |
| 17.5 | 1.5 | 4.65 | \( |1.5 - 4.65| = 3.15 \) |
| 12 | 9 | 4.1 | \( |9 - 4.1| = 4.9 \) |
| 10 | 13 | 3.9 | \( |13 - 3.9| = 9.1 \) |
| 11 | 4 | 4.0 | \( |4 - 4.0| = 0.0 \) |
### Step 3: Sum of Absolute Deviations
Now, we sum up the absolute deviations:
\[
4.0 + 81.05 + 3.15 + 4.9 + 9.1 + 0.0 = 102.2
\]
This means the sum of the absolute deviations is \( 102.2 \).
However, this result seems inconsistent with the options provided in the multiple-choice format. The process I've performed gives the sum of absolute deviations accurately calculated based on the formula.
There may be more context needed, or there could have been a mistake in the given values from the task statement. Nevertheless, per calculations, the sum of absolute deviations would equal \( 102.2 \).
If you're looking for a multiple-choice option, none of the provided options (13.6, 13.62, 24.5, 24.52, 26.46, 72, 26.467) matches the calculated sum of \( 102.2 \). I recommend reviewing to ensure that the data or line equation is accurate, or consider whether or not actual values were correctly transcribed.