Find the correlation coefficient of the line of best fit for the points (−3, –40)

Question

, (1, 12)
, (5, 72)
, and (7, 137)
. Explain how you got your answer. Use the coefficient to describe the correlation of this data.

(4 points)

Answer 1

To find the correlation coefficient (denoted as \( r \)) for the given points \((-3, -40)\), \( (1, 12)\), \( (5, 72)\), and \( (7, 137)\), we will proceed with the following steps:

1. Calculate the means of \( x \) and \( y \): \[ \text{Mean of } x = \frac{-3 + 1 + 5 + 7}{4} = \frac{10}{4} = 2.5 \] \[ \text{Mean of } y = \frac{-40 + 12 + 72 + 137}{4} = \frac{181}{4} = 45.25 \]

2. Calculate the deviations from the means: \[ x_i - \bar{x}, \quad y_i - \bar{y} \]

   For each point:

   • For \((-3, -40)\): \( (-3 - 2.5, -40 - 45.25) = (-5.5, -85.25) \)
   • For \( (1, 12)\): \( (1 - 2.5, 12 - 45.25) = (-1.5, -33.25) \)
   • For \( (5, 72)\): \( (5 - 2.5, 72 - 45.25) = (2.5, 26.75) \)
   • For \( (7, 137)\): \( (7 - 2.5, 137 - 45.25) = (4.5, 91.75) \)

3. Compute the products of the deviations:

   • For \((-3, -40)\): \( (-5.5)(-85.25) = 468.375 \)
   • For \( (1, 12)\): \( (-1.5)(-33.25) = 49.875 \)
   • For \( (5, 72)\): \( (2.5)(26.75) = 66.875 \)
   • For \( (7, 137)\): \( (4.5)(91.75) = 413.875 \)

   Summing these products gives: \[ \sum (x_i - \bar{x})(y_i - \bar{y}) = 468.375 + 49.875 + 66.875 + 413.875 = 999 \]

4. Calculate the sum of squares of deviations: \[ \sum (x_i - \bar{x})^2 = (-5.5)^2 + (-1.5)^2 + (2.5)^2 + (4.5)^2 = 30.25 + 2.25 + 6.25 + 20.25 = 59.00 \] \[ \sum (y_i - \bar{y})^2 = (-85.25)^2 + (-33.25)^2 + (26.75)^2 + (91.75)^2 = 7270.0625 + 1105.0625 + 446.5625 + 8421.0625 = 18842.75 \]

5. Calculate the correlation coefficient \( r \): \[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}} \] Substituting the values we computed: \[ r = \frac{999}{\sqrt{59.00 \cdot 18842.75}} = \frac{999}{\sqrt{1113954.25}} \approx \frac{999}{1055.54} \approx 0.9463 \]

Conclusion

The correlation coefficient \( r \approx 0.9463 \) suggests a strong positive linear relationship between the \( x \) and \( y \) values of the given points. This indicates that as the \( x \) values increase, the \( y \) values tend to increase as well. The closer the coefficient is to 1, the stronger the positive correlation. An \( r \) value above 0.9 indicates a very strong linear correlation in this dataset.