The following table shows the daily distances of a person training to become a runner. Write the line of best fit for the data in form y=mx+b. Round to the nearest hundredth (two decimal places) and do not use spaces in your answer.
Days 1 5 7 10
Miles .21 .28 .34 .45
The line of best fit is...
there are several handy regression calculators online.
Of course, for only four points, it's not too hard by hand.
What do you get?
What is the answer
To find the line of best fit for the given data, we can use the method of least squares. This involves calculating the slope (m) and y-intercept (b) of the line that minimizes the sum of the squared differences between the actual data points and the corresponding predicted values on the line.
To calculate the slope (m):
1. Calculate the mean of the x-values (days) and the mean of the y-values (miles).
Mean of x-values (days):
(1 + 5 + 7 + 10) / 4 = 23 / 4 = 5.75
Mean of y-values (miles):
(.21 + .28 + .34 + .45) / 4 = 1.28 / 4 = 0.32
2. Calculate the differences between each x-value and the mean of x-values (xi - x_mean) and the differences between each y-value and the mean of y-values (yi - y_mean), and multiply them together (xi - x_mean)(yi - y_mean).
(1 - 5.75)(.21 - 0.32) = (-4.75)(-0.11) = 0.5225
(5 - 5.75)(.28 - 0.32) = (-0.75)(-0.04) = 0.03
(7 - 5.75)(.34 - 0.32) = (1.25)(0.02) = 0.025
(10 - 5.75)(.45 - 0.32) = (4.25)(0.13) = 0.5525
3. Calculate the sum of the products from step 2.
0.5225 + 0.03 + 0.025 + 0.5525 = 1.13
4. Calculate the differences between each x-value and the mean of x-values squared (xi - x_mean)^2 and sum them up.
(1 - 5.75)^2 = 22.5625
(5 - 5.75)^2 = 0.5625
(7 - 5.75)^2 = 1.5625
(10 - 5.75)^2 = 18.0625
22.5625 + 0.5625 + 1.5625 + 18.0625 = 42.75
5. Calculate the slope (m) by dividing the sum of the products from step 3 by the sum of the squared differences from step 4.
m = 1.13 / 42.75 = 0.0265 (rounded to four decimal places)
To calculate the y-intercept (b):
1. Substitute the calculated mean values from step 1 and the slope (m) into the equation y = mx + b, and solve for b.
0.32 = 0.0265 * 5.75 + b
2. Simplify the equation and solve for b.
0.32 = 0.151375 + b
b = 0.32 - 0.151375
b = 0.168625 (rounded to six decimal places)
Therefore, the line of best fit is y = 0.0265x + 0.168625.