Given the data set (5, 9), (13, 12), (23, 16), which of the following equations best represents a line of best fit?
y = three fourthsx − 7
y = four fifthsx + 7 ***
y = three over twox + 7
y = two thirdsx − 7
(12-9)/(13-5) = 3/8
(16-12)/(23-13) = 2/5
(16-9)/(23-5) = 7/18
4/5 is nowhere any of those slopes.
The handy calculator at
http://ncalculators.com/statistics/linear-regression-calculator.htm
comes up with
y = .3893x + 7.0123
Looks like none of them is close, since all their slopes are over 1/2. So, the closest is (D).
Thanks so much!
It was wrong it is c
In that case, I suspect the x-y values are reversed.
To determine which equation best represents a line of best fit, we can use the method of least squares. This method involves finding the equation of a line that minimizes the sum of the squared differences between the actual data points and the predicted values on the line.
To calculate the line of best fit, we need to find the slope (m) and y-intercept (b) of the line. The equation of a line is typically written as y = mx + b, where m represents the slope and b represents the y-intercept.
Let's calculate the slope (m) and y-intercept (b) using the given data points:
Data set: (5, 9), (13, 12), (23, 16)
Step 1: Calculate the mean of the x-values (x̄) and the mean of the y-values (ȳ):
x̄ = (5 + 13 + 23) / 3 = 13.67
ȳ = (9 + 12 + 16) / 3 = 12.33
Step 2: Calculate the differences between each x-value (xi) and the mean of the x-values (x̄), and the differences between each y-value (yi) and the mean of the y-values (ȳ). Square these differences:
(x1 - x̄)^2 = (5 - 13.67)^2 = 72.11
(x2 - x̄)^2 = (13 - 13.67)^2 = 0.42
(x3 - x̄)^2 = (23 - 13.67)^2 = 86.07
(y1 - ȳ)^2 = (9 - 12.33)^2 = 11.11
(y2 - ȳ)^2 = (12 - 12.33)^2 = 0.11
(y3 - ȳ)^2 = (16 - 12.33)^2 = 13.47
Step 3: Calculate the sum of the squared differences:
Σ(xi - x̄)^2 = 72.11 + 0.42 + 86.07 ≈ 158.60
Σ(yi - ȳ)^2 = 11.11 + 0.11 + 13.47 ≈ 24.69
Step 4: Calculate the sum of the cross products between the differences of x (xi - x̄) and y (yi - ȳ):
(xi - x̄)(yi - ȳ):
(5 - 13.67)(9 - 12.33) ≈ -8.02
(13 - 13.67)(12 - 12.33) ≈ -0.08
(23 - 13.67)(16 - 12.33) ≈ 12.03
Step 5: Calculate the slope (m):
m = Σ(xi - x̄)(yi - ȳ) / Σ(xi - x̄)^2 ≈ ( -8.02 + (-0.08) + 12.03 ) / 158.60 ≈ 0.06
Step 6: Calculate the y-intercept (b):
b = ȳ - m * x̄ ≈ 12.33 - 0.06 * 13.67 ≈ 11.45
Therefore, the equation that represents the line of best fit is:
y = 0.06x + 11.45
Among the given options, the equation y = four fifthsx + 7 is the closest representation to the line of best fit equation y = 0.06x + 11.45.