Ask a New Question
Menu

Note: The item below has been reviewed and is scheduled to be updated. All students will receive full credit for any response to the following. The least absolute deviation line equation for the data in the table is m = 0.15x + 3 (estimated with technology). x y 1 3 6 2.5 0.5 8 4 1 7.5 1.5 12 9 10 13 11 4 What is the sum of the absolute deviations? (1 point) Responses 10.85 10.85 20.05 20.05 38 38 130.2

1 answer

To calculate the sum of the absolute deviations for the least absolute deviation line equation \( y = 0.15x + 3 \), we first need to find the predicted \( y \) values using the provided equation for each \( x \) in the dataset. Then we will calculate the absolute deviations for each data point and sum them up.

The provided data points are:

x:  1    6    2.5  0.5  8    4    1    7.5  1.5  12   9    10   13   11   4
y:  3    2.5  0.5  8    4    1    7.5  1.5  12   9    10   13   11   4

Now, let's calculate the predicted \( y \) values using the equation \( y = 0.15x + 3 \):

  1. For \( x = 1 \): \( y = 0.15(1) + 3 = 3.15 \)
  2. For \( x = 6 \): \( y = 0.15(6) + 3 = 3.90 \)
  3. For \( x = 2.5 \): \( y = 0.15(2.5) + 3 = 3.375 \)
  4. For \( x = 0.5 \): \( y = 0.15(0.5) + 3 = 3.075 \)
  5. For \( x = 8 \): \( y = 0.15(8) + 3 = 4.20 \)
  6. For \( x = 4 \): \( y = 0.15(4) + 3 = 3.60 \)
  7. For \( x = 1 \): \( y = 0.15(1) + 3 = 3.15 \)
  8. For \( x = 7.5 \): \( y = 0.15(7.5) + 3 = 4.125 \)
  9. For \( x = 1.5 \): \( y = 0.15(1.5) + 3 = 3.225 \)
  10. For \( x = 12 \): \( y = 0.15(12) + 3 = 4.80 \)
  11. For \( x = 9 \): \( y = 0.15(9) + 3 = 4.35 \)
  12. For \( x = 10 \): \( y = 0.15(10) + 3 = 4.50 \)
  13. For \( x = 13 \): \( y = 0.15(13) + 3 = 4.95 \)
  14. For \( x = 11 \): \( y = 0.15(11) + 3 = 4.80 \)
  15. For \( x = 4 \): \( y = 0.15(4) + 3 = 3.60 \)

Now we can calculate the absolute deviations for each of the data points:

  1. \( |3 - 3.15| = 0.15 \)
  2. \( |2.5 - 3.90| = 1.40 \)
  3. \( |0.5 - 3.375| = 2.875 \)
  4. \( |8 - 3.075| = 4.925 \)
  5. \( |4 - 4.20| = 0.20 \)
  6. \( |1 - 3.60| = 2.60 \)
  7. \( |7.5 - 3.15| = 4.35 \)
  8. \( |1.5 - 4.125| = 2.625 \)
  9. \( |12 - 3.225| = 8.775 \)
  10. \( |9 - 4.35| = 4.65 \)
  11. \( |10 - 4.50| = 5.50 \)
  12. \( |13 - 4.95| = 8.05 \)
  13. \( |4 - 4.80| = 0.80 \)
  14. \( |4 - 3.60| = 0.40 \)

Now we sum all the absolute deviations:

\[ \text{Sum of Absolute Deviations} = 0.15 + 1.40 + 2.875 + 4.925 + 0.20 + 2.60 + 4.35 + 2.625 + 8.775 + 4.65 + 5.50 + 8.05 + 0.80 + 0.40 \]

Calculating the total:

\[ = 0.15 + 1.40 + 2.875 + 4.925 + 0.20 + 2.60 + 4.35 + 2.625 + 8.775 + 4.65 + 5.50 + 8.05 + 0.80 + 0.40 = 43.45 \]

So the total sum of absolute deviations is \( \approx 43.45 \).

Looking at the provided choices, it seems that there could be either an error in my calculations or an issue with the choices. Based on my computation of the absolute deviations, the answer is not available among the options presented. Please double-check the values in the table or clarify the conditions of the problem. The correct response should match calculation outputs directly.