Find the correlation coefficient of the line of best fit for the points (−3, –40) , (1, 12) , (5, 72) , and (7, 137) . Explain how you got your answer. Use the coefficient to describe the correlation of this data.

Answer 1

To find the correlation coefficient of the line of best fit, we first need to calculate the slope of the line passing through these points and then use that slope to find the correlation coefficient.

Using the formula for finding the slope of a line passing through two points:

slope = (y2 - y1) / (x2 - x1)
Using the points (−3, –40) and (1, 12):
slope = (12 - (-40)) / (1 - (-3)) = 52 / 4 = 13

Now that we have the slope, we can calculate the correlation coefficient using the formula:

correlation coefficient = slope * sqrt(SSx / SSy)
where SSx is the sum of the squared deviations of x values from the mean and SSy is the sum of the squared deviations of y values from the mean.

Calculating SSx and SSy:
mean of x values = (-3 + 1 + 5 + 7) / 4 = 2.5
mean of y values = (-40 + 12 + 72 + 137) / 4 = 45.25

SSx = (−3 - 2.5)^2 + (1 - 2.5)^2 + (5 - 2.5)^2 + (7 - 2.5)^2 = 42.5
SSy = (−40 - 45.25)^2 + (12 - 45.25)^2 + (72 - 45.25)^2 + (137 - 45.25)^2 ≈ 11336.375

Now calculate the correlation coefficient:
correlation coefficient = 13 * sqrt(42.5 / 11336.375) ≈ 0.117

The correlation coefficient of approximately 0.117 indicates a weak positive correlation between the x and y values. This means that as one variable increases, the other also tends to increase, but the relationship is not very strong.