To find the equation of the line that best fits the data using linear regression, we can follow these steps:
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Calculate the means of x and y: \[ \bar{x} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5 \] \[ \bar{y} = \frac{68 + 79 + 97 + 115 + 130 + 140}{6} = \frac{629}{6} \approx 104.8333 \]
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Calculate the slope (m): The formula for the slope \(m\) is: \[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
First, we calculate \( \sum (x_i - \bar{x})(y_i - \bar{y}) \) and \( \sum (x_i - \bar{x})^2 \):
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For each \(x_i\) and \(y_i\):
- \( (x_1, y_1) = (1, 68) \)
- \( (x_2, y_2) = (2, 79) \)
- \( (x_3, y_3) = (3, 97) \)
- \( (x_4, y_4) = (4, 115) \)
- \( (x_5, y_5) = (5, 130) \)
- \( (x_6, y_6) = (6, 140) \)
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Calculate the deviations:
- \( x_i - \bar{x} = [-2.5, -1.5, -0.5, 0.5, 1.5, 2.5] \)
- \( y_i - \bar{y} = [-36.83, -25.83, -7.83, 10.17, 25.17, 35.17] \)
Now we compute: \[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (-2.5)(-36.83) + (-1.5)(-25.83) + (-0.5)(-7.83) + (0.5)(10.17) + (1.5)(25.17) + (2.5)(35.17) \] \[ = 95.825 + 38.745 + 3.915 + 5.085 + 37.755 + 87.925 = 269.2 \]
For the denominator: \[ \sum (x_i - \bar{x})^2 = (-2.5)^2 + (-1.5)^2 + (-0.5)^2 + (0.5)^2 + (1.5)^2 + (2.5)^2 \] \[ = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5 \]
Now we can calculate the slope \(m\): \[ m = \frac{269.2}{17.5} \approx 15.36 \]
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Calculate the y-intercept (b): The formula for the y-intercept \(b\) is: \[ b = \bar{y} - m \bar{x} \] \[ b = 104.8333 - (15.36)(3.5) \approx 104.8333 - 53.76 = 51.0733 \approx 51.07 \]
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Final Equation: Now, substituting \(m\) and \(b\) into the equation of a line \(y = mx + b\): \[ y \approx 15.36x + 51.07 \]
Thus, the equation of the line that best fits the data is:
\[ y = 15.36x + 51.07 \]
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