To find the slope of the least squares regression line from the provided data points, we can use the formula for the slope \( m \):
\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
Where:
- \( N \) is the number of points (in this case, 5)
- \( \sum xy \) is the sum of the product of each \( x \) and \( y \)
- \( \sum x \) is the sum of the \( x \) values
- \( \sum y \) is the sum of the \( y \) values
- \( \sum x^2 \) is the sum of the square of each \( x \)
Now, let's calculate the needed values:
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List the points:
\[ (0, 12), (1, 14), (2, 16), (3, 18), (4, 20) \]
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Calculate \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \):
- \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
- \( \sum y = 12 + 14 + 16 + 18 + 20 = 80 \)
- \( \sum xy = (0 \cdot 12) + (1 \cdot 14) + (2 \cdot 16) + (3 \cdot 18) + (4 \cdot 20) = 0 + 14 + 32 + 54 + 80 = 180 \)
- \( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30 \)
Now we have:
- \( N = 5 \)
- \( \sum x = 10 \)
- \( \sum y = 80 \)
- \( \sum xy = 180 \)
- \( \sum x^2 = 30 \)
- Plug values into the slope formula:
\[ m = \frac{5(180) - (10)(80)}{5(30) - (10)^2} \]
Calculating the numerator:
\[ 5(180) = 900 \] \[ (10)(80) = 800 \] So, \[ 900 - 800 = 100 \]
Calculating the denominator:
\[ 5(30) = 150 \] \[ (10)^2 = 100 \] So, \[ 150 - 100 = 50 \]
Finally, calculate the slope:
\[ m = \frac{100}{50} = 2 \]
Thus, the slope of the least squares regression line is 2.
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