The table shows the age of several people and how many hours a day they spend playing video games.
Age in Years (x) 15 60 30 20 10 40
Days Playing (y) 5 0 2 8 8 4Which equation could represent a line of best fit for this data?
y ≈ 8.78x + 15
y ≈ 0.15x + 60
y ≈ −0.15x + 8.78
y ≈ −1.5x + 60
To determine the equation that represents the line of best fit for this data, you can use linear regression. Linear regression finds the best line that fits the scatterplot of the data points.
Step 1: Calculate the mean (average) of both the x-values (ages) and the y-values (days playing).
Mean of x: (15 + 60 + 30 + 20 + 10 + 40) / 6 = 175 / 6 ≈ 29.17
Mean of y: (5 + 0 + 2 + 8 + 8 + 4) / 6 = 27 / 6 = 4.5
Step 2: Subtract the mean from each x-value and y-value to calculate the deviations.
Age deviations: 15 - 29.17 = -14.17, 60 - 29.17 = 30.83, 30 - 29.17 = 0.83, 20 - 29.17 = -9.17, 10 - 29.17 = -19.17, 40 - 29.17 = 10.83
Days playing deviations: 5 - 4.5 = 0.5, 0 - 4.5 = -4.5, 2 - 4.5 = -2.5, 8 - 4.5 = 3.5, 8 - 4.5 = 3.5, 4 - 4.5 = -0.5
Step 3: Multiply the deviations for each data point to calculate the products.
Product 1: (-14.17)(0.5) = -7.085
Product 2: (30.83)(-4.5) = -138.735
Product 3: (0.83)(-2.5) = -2.075
Product 4: (-9.17)(3.5) = -32.095
Product 5: (-19.17)(3.5) = -67.095
Product 6: (10.83)(-0.5) = -5.415
Step 4: Calculate the sum of the products.
Sum of the products: -7.085 + (-138.735) + (-2.075) + (-32.095) + (-67.095) + (-5.415) = -252.5
Step 5: Calculate the sum of the squared deviations for x.
Deviation 1 squared: (-14.17)^2 = 200.8489
Deviation 2 squared: (30.83)^2 = 950.4889
Deviation 3 squared: (0.83)^2 = 0.6889
Deviation 4 squared: (-9.17)^2 = 84.0489
Deviation 5 squared: (-19.17)^2 = 367.2489
Deviation 6 squared: (10.83)^2 = 117.3689
Sum of squared deviations: 200.8489 + 950.4889 + 0.6889 + 84.0489 + 367.2489 + 117.3689 = 1720.6824
Step 6: Calculate the slope (b) of the line of best fit.
b = sum of the products / sum of squared deviations
b = -252.5 / 1720.6824 ≈ -0.146729
Step 7: Calculate the y-intercept (a) of the line of best fit.
a = mean of y - b * mean of x
a = 4.5 - (-0.146729 * 29.17) ≈ 8.78581
Step 8: Write the equation of the line in y = mx + b form.
y ≈ -0.146729x + 8.78581
Therefore, the equation that represents the line of best fit for this data is approximately y ≈ -0.146729x + 8.78581.
Given the options, the closest equation to this line is y ≈ -0.15x + 8.78.