To find the correlation coefficient \( r \) of the line of best fit for the given points \( (-3, -40) \), \( (1, 12) \), \( (5, 72) \), and \( (7, 137) \), we can follow these steps:
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Calculate means: First, we need to calculate the means of the x-values and y-values.
\[ \text{Mean of } x = \frac{-3 + 1 + 5 + 7}{4} = \frac{10}{4} = 2.5 \] \[ \text{Mean of } y = \frac{-40 + 12 + 72 + 137}{4} = \frac{181}{4} = 45.25 \]
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Calculate sums needed for covariance and variance: We need to calculate \( S_{xy} \) (covariance) and \( S_{xx} \) (variance of x).
\[ S_{xy} = \sum{(x_i - \bar{x})(y_i - \bar{y})} \] \[ S_{xx} = \sum{(x_i - \bar{x})^2} \]
Now, for each point, compute \( (x_i - \bar{x}) \), \( (y_i - \bar{y}) \), and the products:
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For \( (-3, -40) \): \[ x_i - \bar{x} = -3 - 2.5 = -5.5, \quad y_i - \bar{y} = -40 - 45.25 = -85.25 \] \[ (x_i - \bar{x})(y_i - \bar{y}) = (-5.5)(-85.25) = 468.375 \] \[ (x_i - \bar{x})^2 = (-5.5)^2 = 30.25 \]
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For \( (1, 12) \): \[ x_i - \bar{x} = 1 - 2.5 = -1.5, \quad y_i - \bar{y} = 12 - 45.25 = -33.25 \] \[ (x_i - \bar{x})(y_i - \bar{y}) = (-1.5)(-33.25) = 49.875 \] \[ (x_i - \bar{x})^2 = (-1.5)^2 = 2.25 \]
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For \( (5, 72) \): \[ x_i - \bar{x} = 5 - 2.5 = 2.5, \quad y_i - \bar{y} = 72 - 45.25 = 26.75 \] \[ (x_i - \bar{x})(y_i - \bar{y}) = (2.5)(26.75) = 66.875 \] \[ (x_i - \bar{x})^2 = (2.5)^2 = 6.25 \]
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For \( (7, 137) \): \[ x_i - \bar{x} = 7 - 2.5 = 4.5, \quad y_i - \bar{y} = 137 - 45.25 = 91.75 \] \[ (x_i - \bar{x})(y_i - \bar{y}) = (4.5)(91.75) = 413.875 \] \[ (x_i - \bar{x})^2 = (4.5)^2 = 20.25 \]
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Sum up the products:
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Sum of \( S_{xy} \): \[ S_{xy} = 468.375 + 49.875 + 66.875 + 413.875 = 999 \]
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Sum of \( S_{xx} \): \[ S_{xx} = 30.25 + 2.25 + 6.25 + 20.25 = 59.00 \]
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Calculate the correlation coefficient \( r \): The correlation coefficient is given by: \[ r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}} \] First, we need to calculate \( S_{yy} \): \[ S_{yy} = \sum{(y_i - \bar{y})^2} \]
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For \( (-40) \): \[ (y_i - \bar{y})^2 = (-85.25)^2 = 7268.0625 \]
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For \( (12) \): \[ (y_i - \bar{y})^2 = (-33.25)^2 = 1105.5625 \]
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For \( (72) \): \[ (y_i - \bar{y})^2 = (26.75)^2 = 447.5625 \]
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For \( (137) \): \[ (y_i - \bar{y})^2 = (91.75)^2 = 8421.0625 \]
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Sum up to get \( S_{yy} \): \[ S_{yy} = 7268.0625 + 1105.5625 + 447.5625 + 8421.0625 = 16842.25 \]
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Now substitute back to find \( r \): \[ r = \frac{999}{\sqrt{59.00 \times 16842.25}} \approx \frac{999}{\sqrt{992417.25}} \approx \frac{999}{996.206565} \approx 1.003 \]
Using the correlation coefficient \( r \approx 1.003 \), which is around 1, we can conclude that there is a very strong positive correlation between the x and y values in this dataset.
In summary, the steps involved calculating the means, products for covariance and variance, then using these to compute the correlation coefficient \( r \). Since its value is close to 1, it indicates a strong positive linear relationship between the variables.