Find the line \( y=a+b x \) which best approximates the data points \[ (-4,63),(-3,44),(-1,11),(0,-6),(2,-46) \] \( y= \)

To find the line \( y = a + b x \) that best approximates the given points using the method of least squares, we need to determine the coefficients \( a \) (the intercept) and \( b \) (the slope). The given data points are: \[ (-4, 63), (-3, 44), (-1, 11), (0, -6), (2, -46) \] We can follow these steps: 1. **Calculate the means:** \[ \bar{x} = \frac{-4 + -3 + -1 + 0 + 2}{5} = \frac{-6}{5} = -1.2 \] \[ \bar{y} = \frac{63 + 44 + 11 - 6 - 46}{5} = \frac{66}{5} = 13.2 \] 2. **Calculate the slope \( b \):** \[ b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \] Let's compute the components: \[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (-4 + 1.2)(63 - 13.2) + (-3 + 1.2)(44 - 13.2) + (-1 + 1.2)(11 - 13.2) + (0 + 1.2)(-6 - 13.2) + (2 + 1.2)(-46 - 13.2) \] \[ = (-2.8)(49.8) + (-1.8)(30.8) + (0.2)(-2.2) + (1.2)(-19.2) + (3.2)(-59.2) \] \[ = -139.44 - 55.44 - 0.44 - 23.04 - 189.44 = -407.8 \] Next, calculate: \[ \sum (x_i - \bar{x})^2 = (-4 + 1.2)^2 + (-3 + 1.2)^2 + (-1 + 1.2)^2 + (0 + 1.2)^2 + (2 + 1.2)^2 \] \[ = (-2.8)^2 + (-1.8)^2 + (0.2)^2 + (1.2)^2 + (3.2)^2 \] \[ = 7.84 + 3.24 + 0.04 + 1.44 + 10.24 = 22.8 \] Now we can find \( b \): \[ b = \frac{-407.8}{22.8} \approx -17.9 \] 3. **Calculate the intercept \( a \):** \[ a = \bar{y} - b\bar{x} \] \[ a = 13.2 - (-17.9)(-1.2) \] \[ = 13.2 - 21.48 = -8.28 \] Finally, the line that best approximates the data points is: \[ y \approx -8.28 - 17.9x \]