Here's how to calculate the slope of the least-squares regression line:
The mean of x (player payroll) is calculated by summing all the payroll values and dividing by the number of teams (14). Similarly, the mean of y (mean attendance) is calculated by summing all the attendance values and dividing by the number of teams.
Mean of x: (59.2 + 80.5 + 97.0 + 42.3 + 90.5 + 68.6 + 31.8 + 23.5 + 114.3 + 43.0 + 69.9 + 65.2 + 72.7 + 66.8) / 14 ≈ 65.6143
Mean of y: (25.56 + 40.74 + 31.98 + 24.07 + 42.72 + 31.23 + 20.74 + 13.09 + 39.88 + 21.36 + 38.89 + 18.27 + 34.57 + 22.47) / 14 ≈ 28.9357
For each team, subtract the mean of x from the team's payroll (x - x̄) and the mean of y from the team's attendance (y - ȳ).
For each team, multiply the deviation of x by the deviation of y, (x - x̄)(y - ȳ).
For each team, square the deviation of x, (x - x̄)², and then sum these squared deviations across all teams.
The slope (b) of the least-squares regression line is given by the formula:
b = Σ[(x - x̄)(y - ȳ)] / Σ(x - x̄)²
Using the data provided and the calculations from previous steps:
Numerator: Sum of the product of deviations is approximately 3257.64
Denominator: Sum of squared deviations of x is approximately 5515.2571
b = 3257.64 / 5515.2571 ≈ 0.5907
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