Answer:
To find the equation of the line of best fit, we can use linear regression. Linear regression helps us determine the equation of a straight line that best fits a given set of data points. In this case, we have the following data:
X = [1, 3, 5, 9, 16]
y = [20, 14, 10, 6, 4]
Let's calculate the slope (m) and y-intercept (b) of the line of best fit using linear regression:
Step 1: Calculate the means of X and y.
Mean of X: (1 + 3 + 5 + 9 + 16) / 5 = 6.8
Mean of y: (20 + 14 + 10 + 6 + 4) / 5 = 10.8
Step 2: Calculate the deviations from the means.
Deviations from the mean of X: [1 - 6.8, 3 - 6.8, 5 - 6.8, 9 - 6.8, 16 - 6.8] = [-5.8, -3.8, -1.8, 2.2, 9.2]
Deviations from the mean of y: [20 - 10.8, 14 - 10.8, 10 - 10.8, 6 - 10.8, 4 - 10.8] = [9.2, 3.2, -0.8, -4.8, -6.8]
Step 3: Calculate the sum of the products of the deviations.
Sum of the products of deviations: (-5.8 * 9.2) + (-3.8 * 3.2) + (-1.8 * -0.8) + (2.2 * -4.8) + (9.2 * -6.8) = -62.08
Step 4: Calculate the sum of the squared deviations of X.
Sum of squared deviations of X: (-5.8^2) + (-3.8^2) + (-1.8^2) + (2.2^2) + (9.2^2) = 175.8
Step 5: Calculate the slope (m) of the line of best fit.
Slope (m) = Sum of the products of deviations / Sum of squared deviations of X = -62.08 / 175.8 ≈ -0.352
Step 6: Calculate the y-intercept (b) of the line of best fit.
y-intercept (b) = Mean of y - (slope * Mean of X) = 10.8 - (-0.352 * 6.8) ≈ 13.536
Therefore, the equation of the line of best fit, rounded to three decimal places, is:
y = -0.352x + 13.536
Explanation:
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