To find the slope of the least squares regression line for the given 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,
- \( \sum xy \) is the sum of the product of each \( x \) and \( y \),
- \( \sum x \) is the sum of all \( x \) values,
- \( \sum y \) is the sum of all \( y \) values,
- \( \sum x^2 \) is the sum of the squares of each \( x \) value.
Given your data points:
- \( (0, 12) \)
- \( (1, 14) \)
- \( (2, 16) \)
- \( (3, 18) \)
- \( (4, 20) \)
We first calculate the required sums:
- Calculating \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \):
\[ \sum x = 0 + 1 + 2 + 3 + 4 = 10 \]
\[ \sum y = 12 + 14 + 16 + 18 + 20 = 80 \]
\[ \sum xy = (0 \cdot 12) + (1 \cdot 14) + (2 \cdot 16) + (3 \cdot 18) + (4 \cdot 20) = 0 + 14 + 32 + 54 + 80 = 180 \]
\[ \sum x^2 = (0^2) + (1^2) + (2^2) + (3^2) + (4^2) = 0 + 1 + 4 + 9 + 16 = 30 \]
- Calculating \( N \):
\[ N = 5 \]
- Now plug into the slope formula:
\[ m = \frac{5(180) - (10)(80)}{5(30) - (10)^2} \] \[ m = \frac{900 - 800}{150 - 100} \] \[ m = \frac{100}{50} = 2 \]
Therefore, the slope of the least squares regression line based on the given points is \( \boxed{2} \).