To find the slope of the least-squares regression line for the given dataset, we can use the formula for the slope \( b \) of the regression line:
\[ b = \frac{n(\sum{xy}) - (\sum{x})(\sum{y})}{n(\sum{x^2}) - (\sum{x})^2} \]
Where:
- \( n \) = number of data points
- \( \sum{xy} \) = sum of the product of each pair of \( x \) and \( y \)
- \( \sum{x} \) = sum of the \( x \) values
- \( \sum{y} \) = sum of the \( y \) values
- \( \sum{x^2} \) = sum of the squares of the \( x \) values
Step 1: Organize the Data
We first extract the data from the table.
\[ \begin{align*} (x, y) \ (1, 15) \ (3, 17) \ (6, 18) \ (7, 18) \ (9, 20) \ (15, 24) \ (16, 23) \ (22, 26) \ (23, 27) \ (28, 30) \ (33, 32) \end{align*} \]
Step 2: Calculate the necessary summations
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Calculate \( n \) (the number of points):
- \( n = 11 \)
-
Calculate \( \sum{x} \):
- \( \sum{x} = 1 + 3 + 6 + 7 + 9 + 15 + 16 + 22 + 23 + 28 + 33 = 163 \)
-
Calculate \( \sum{y} \):
- \( \sum{y} = 15 + 17 + 18 + 18 + 20 + 24 + 23 + 26 + 27 + 30 + 32 = 239 \)
-
Calculate \( \sum{xy} \):
- \( \sum{xy} = 1 \cdot 15 + 3 \cdot 17 + 6 \cdot 18 + 7 \cdot 18 + 9 \cdot 20 + 15 \cdot 24 + 16 \cdot 23 + 22 \cdot 26 + 23 \cdot 27 + 28 \cdot 30 + 33 \cdot 32 \)
- \( = 15 + 51 + 108 + 126 + 180 + 360 + 368 + 572 + 621 + 840 + 1056 = 3227 \)
-
Calculate \( \sum{x^2} \):
- \( \sum{x^2} = 1^2 + 3^2 + 6^2 + 7^2 + 9^2 + 15^2 + 16^2 + 22^2 + 23^2 + 28^2 + 33^2 \)
- \( = 1 + 9 + 36 + 49 + 81 + 225 + 256 + 484 + 529 + 784 + 1089 = 2259 \)
Step 3: Substitute into the slope formula
Now we can substitute our values into the slope formula:
\[ b = \frac{11(3227) - (163)(239)}{11(2259) - (163)^2} \]
Calculate the terms:
Numerator: \[ 11(3227) = 35497 \] \[ (163)(239) = 38957 \] \[ \text{Numerator} = 35497 - 38957 = -3440 \]
Denominator: \[ 11(2259) = 24849 \] \[ (163)^2 = 26569 \] \[ \text{Denominator} = 24849 - 26569 = -1720 \]
Thus, the slope \( b \) becomes:
\[ b = \frac{-3440}{-1720} \approx 2 \]
Conclusion
Based on the above calculations, the slope of the least-squares regression line is approximately \( 2 \). However, that means none of the provided options seem to match closely. Would you like me to recalculate or check any assumptions?