To calculate the slope of the least squares regression line based on the given points, we can use the formula for the slope \( m \) of the regression line:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]
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, and
- \( \sum x^2 \) is the sum of the squares of each x value.
First, calculate the necessary sums from the table provided:
| x | y | xy | x² | |---|----|----|----| | 0 | 12 | 0 | 0 | | 1 | 14 | 14 | 1 | | 2 | 16 | 32 | 4 | | 3 | 18 | 54 | 9 | | 4 | 20 | 80 | 16 |
Now we compute the sums:
- \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
- \( \sum y = 12 + 14 + 16 + 18 + 20 = 80 \)
- \( \sum xy = 0 + 14 + 32 + 54 + 80 = 180 \)
- \( \sum x^2 = 0 + 1 + 4 + 9 + 16 = 30 \)
Now plug these values into the slope formula:
- \( n = 5 \)
\[ m = \frac{5(180) - (10)(80)}{5(30) - (10)^2} \] \[ m = \frac{900 - 800}{150 - 100} \] \[ m = \frac{100}{50} = 2 \]
Thus, the slope of the least squares regression line is \( \boxed{2} \).
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