Using the data from question #1, what is the equation of the line of best fit? (For future reference: line of best fit, linear regression equation, and least-squares line are all the same thing y=ax+b). Round your answers to the nearest tenth.
x : {20,30,45,60,80,90,45,120,90,70} y: {50,45,55,70,80,90,80,100,95,85}
To find the equation of the line of best fit, we need to perform linear regression on the given data. Here are the steps:
1. Calculate the means of x and y:
- mean of x (x-bar) = (20 + 30 + 45 + 60 + 80 + 90 + 45 + 120 + 90 + 70) / 10 = 70
- mean of y (y-bar) = (50 + 45 + 55 + 70 + 80 + 90 + 80 + 100 + 95 + 85) / 10 = 78
2. Calculate the deviations from the means for every x and y value:
- deviation of x (x-x-bar): [-50, -40, -25, -10, 10, 20, -25, 50, 20, 0]
- deviation of y (y-y-bar): [-28, -33, -23, -8, 2, 12, 2, 22, 17, 7]
3. Calculate the sum of the products of deviations (x-x-bar) * (y-y-bar), and the sum of the squared deviations of x (x-x-bar)^2:
- sum of products of deviations = (-50 * -28) + (-40 * -33) + (-25 * -23) + (-10 * -8) + (10 * 2) + (20 * 12) + (-25 * 2) + (50 * 22) + (20 * 17) + (0 * 7) = 7600
- sum of squared deviations of x = (-50)^2 + (-40)^2 + (-25)^2 + (-10)^2 + (10)^2 + (20)^2 + (-25)^2 + (50)^2 + (20)^2 + (0)^2 = 20,900
4. Calculate the slope (a) of the line of best fit:
- slope (a) = sum of products of deviations / sum of squared deviations of x = 7600 / 20,900 = 0.3635
5. Calculate the y-intercept (b) of the line of best fit:
- y-intercept (b) = y-bar - (a * x-bar) = 78 - (0.3635 * 70) = 52.4565
6. Write the equation of the line of best fit:
- y = ax + b
- y = 0.3635x + 52.4565
Rounding to the nearest tenth, the equation of the line of best fit is y = 0.4x + 52.5.
To find the equation of the line of best fit (also known as the linear regression equation or the least-squares line), we need to calculate the slope (a) and the y-intercept (b) using the given data points.
Step 1: Calculate the mean (average) of both x and y values.
x: {20, 30, 45, 60, 80, 90, 45, 120, 90, 70}
y: {50, 45, 55, 70, 80, 90, 80, 100, 95, 85}
Mean of x = (20 + 30 + 45 + 60 + 80 + 90 + 45 + 120 + 90 + 70) / 10 = 67
Mean of y = (50 + 45 + 55 + 70 + 80 + 90 + 80 + 100 + 95 + 85) / 10 = 76
Step 2: Calculate the differences between each x value and the mean of x (67), and the differences between each y value and the mean of y (76).
(x - mean of x): {-47, -37, -22, -7, 13, 23, -22, 53, 23, 3}
(y - mean of y): {-26, -31, -21, -6, 4, 14, 4, 24, 19, 9}
Step 3: Calculate the sum of the product of the differences obtained in step 2.
Sum of ((x - mean of x) * (y - mean of y)): 4346
Step 4: Calculate the sum of the squared differences between each x value and the mean of x.
Sum of ((x - mean of x)^2): 50580
Step 5: Calculate the slope (a) using the formula:
a = Sum of ((x - mean of x) * (y - mean of y)) / Sum of ((x - mean of x)^2)
a = 4346 / 50580 ≈ 0.086
Step 6: Calculate the y-intercept (b) using the formula:
b = mean of y - (a * mean of x)
b = 76 - (0.086 * 67) ≈ 70.4
Therefore, the equation of the line of best fit is:
y ≈ 0.1x + 70.4 (rounded to the nearest tenth)