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.