To find the least squares regression line (y = mx + b), we first need to calculate the slope (m) and the y-intercept (b) using the formulae:
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Calculate the slope (m): \[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
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Calculate the y-intercept (b): \[ b = \frac{\sum y - m(\sum x)}{N} \]
Where:
- N is the number of points
- \(\sum xy\) is the sum of the products of x and y
- \(\sum x\) is the sum of x values
- \(\sum y\) is the sum of y values
- \(\sum x^2\) is the sum of the squares of x values
Let's perform the calculations using the provided points in the table.
Data Points:
- \( (-5, 3) \)
- \( (-4, 4) \)
- \( (-3, 7) \)
- \( (-2, 7.5) \)
- \( (-1, 2.5) \)
Calculations:
- \(N = 5\) (number of data points)
- \(\sum x = -5 + (-4) + (-3) + (-2) + (-1) = -15\)
- \(\sum y = 3 + 4 + 7 + 7.5 + 2.5 = 24\)
- \(\sum xy = (-5)(3) + (-4)(4) + (-3)(7) + (-2)(7.5) + (-1)(2.5) = -15 - 16 - 21 - 15 + (-2.5) = -69.5\)
- \(\sum x^2 = (-5)^2 + (-4)^2 + (-3)^2 + (-2)^2 + (-1)^2 = 25 + 16 + 9 + 4 + 1 = 55\)
Now, substituting into the formulas:
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Calculate m: \[ m = \frac{5(-69.5) - (-15)(24)}{5(55) - (-15)^2} \] \[ m = \frac{-347.5 + 360}{275 - 225} \] \[ m = \frac{12.5}{50} = 0.25 \]
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Calculate b: \[ b = \frac{24 - 0.25(-15)}{5} \] \[ b = \frac{24 + 3.75}{5} = \frac{27.75}{5} = 5.55 \]
Final Equation:
Putting it all together, the equation of the least squares regression line is: \[ y = 0.25x + 5.55 \]
Among the response options:
- y = 0.25x + 5.55 is the correct answer.