To find the slope of the least squares regression line using the points provided in the table, 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 data points, \( \sum xy \) is the sum of the product of each \( x \) and \( y \), \( \sum x \) is the sum of \( x \) values, \( \sum y \) is the sum of \( y \) values, and \( \sum x^2 \) is the sum of squared \( x \) values.
Using the data from the table:
\[ \begin{align*} x & : 0, 1, 2, 3, 4 \ y & : 12, 14, 16, 18, 20 \end{align*} \]
We can calculate the required sums:
- \( n = 5 \)
- \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
- \( \sum y = 12 + 14 + 16 + 18 + 20 = 80 \)
- \( \sum xy = (0 \cdot 12) + (1 \cdot 14) + (2 \cdot 16) + (3 \cdot 18) + (4 \cdot 20) = 0 + 14 + 32 + 54 + 80 = 180 \)
- \( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30 \)
Now, plugging these values into the slope formula:
\[ m = \frac{5(180) - (10)(80)}{5(30) - (10)^2} \]
Calculating the numerator:
\[ 5(180) = 900 \] \[ (10)(80) = 800 \] \[ 900 - 800 = 100 \]
Calculating the denominator:
\[ 5(30) = 150 \] \[ (10)^2 = 100 \] \[ 150 - 100 = 50 \]
Now, substituting values into the formula for \( m \):
\[ m = \frac{100}{50} = 2 \]
Thus, the slope of the least squares regression line is \( \boxed{2} \).
1 answer