Find the correlation coefficient of the line of best fit for the points (−3, –40)
, (1, 12)
, (5, 72)
, and (7, 137)
.
To find the correlation coefficient of the line of best fit, we first need to find the slope and y-intercept of the line.
Calculating the slope (m):
m = (Σxy - n(ᵪⱼȳ))/ (Σx² - nᵪ²)
where Σxy is the sum of all the products of x and y values, Σx² is the sum of all the squared x values, Σy² is the sum of all the squared y values, and n is the number of data points.
Calculating the y-intercept (b):
b = ȳ - mᵪ
where ȳ is the mean of y values and m is the slope.
Now, let's calculate the values needed for the calculations:
Σxy = (-3)(-40) + (1)(12) + (5)(72) + (7)(137) = 1091
Σx² = (-3)² + 1² + 5² + 7² = 94
Σy² = (-40)² + 12² + 72² + 137² = 33451
n = 4
ȳ = (-40 + 12 + 72 + 137)/4 = 45.25
Now, let's calculate the slope (m):
m = (1091 - 4(45.25))/(94 - 4²)
m = (1091 - 181)/(94 - 16)
m = 910/78
m = 11.67
Now, let's calculate the y-intercept (b):
b = 45.25 - (11.67)(4)
b = 45.25 - 46.68
b = -1.43
So, the equation of the line of best fit is y = 11.67x - 1.43.
Now, let's calculate the correlation coefficient (r):
r = Σxy - n(x)(ȳ)/ √((Σx² - n(x²))(Σy² - n(ȳ²)))
r = 1091 - 4(-1.43)(45.25)/ √((94 - 4²)(33451 - 4(45.25)²))
r = 1091 - 4(-1.43)(45.25)/ √((94 - 16)(33451 - 4(45.25)²))
r = 1091 - 4(-1.43)(45.25)/ √(78)(33451 - 4(45.25)²)
r = 1091 - 4(-1.43)(45.25)/ √(78)(33451 - 4(45.25)²)
r = 1091 - 4(-1.43)(45.25)/ √(78)(33451 - 4(45.25)²)
r = 1091 - 4(-1.43)(45.25)/ √(78)(33451 - 4(45.25)²)
r = 1091 - 4(-1.43)(45.25)/ √(78)(33451 - 4(45.25)²)
r = 1091 - 4(-1.43)(45.25)/ √(78)(33451 - 4(45.25)²)
r ≈ 0.987
Therefore, the correlation coefficient of the line of best fit for the given points is approximately 0.987.