To determine the line of best fit for the given data, we can use the method of linear regression. The data points provided are:
\[ \begin{align*} (0, 0) \ (1, 712) \ (2, 19) \ (3, 25) \ (4, 0) \end{align*} \]
This gives us the following set of values:
- \(n = 5\) (number of points)
- \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
- \( \sum y = 0 + 712 + 19 + 25 + 0 = 756 \)
- \( \sum xy = (0 \cdot 0) + (1 \cdot 712) + (2 \cdot 19) + (3 \cdot 25) + (4 \cdot 0) = 0 + 712 + 38 + 75 + 0 = 825 \)
- \( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30 \)
We can use the formulas for the slope \(m\) and the y-intercept \(b\) of the line of best fit:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]
\[ b = \frac{\sum y - m(\sum x)}{n} \]
Now we can compute these values:
- Calculate \(m\):
\[ m = \frac{5(825) - (10)(756)}{5(30) - (10)^2} = \frac{4125 - 7560}{150 - 100} = \frac{-3435}{50} = -68.7 \]
- Calculate \(b\):
\[ b = \frac{756 - (-68.7)(10)}{5} = \frac{756 + 687}{5} = \frac{1443}{5} = 288.6 \]
Putting it all together, the equation of the line of best fit is:
\[ y = -68.7x + 288.6 \]
Rounding the coefficients to the nearest tenth, we find:
\[ \boxed{y = -68.7x + 288.6} \]
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