To find the equation for the least squares regression line \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, we need to calculate the values of \( m \) and \( b \) using the provided data. Let’s calculate it step by step:
Step 1: Calculate necessary sums
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Data points: \[ (x_1, y_1) = (2292, 149) \ (x_2, y_2) = (7315, 339) \ (x_3, y_3) = (11518, 357) \ (x_4, y_4) = (13977, 445) \ (x_5, y_5) = (19877, 380) \]
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Calculate sums: \[ n = 5 \ \sum x = 2292 + 7315 + 11518 + 13977 + 19877 = 43679 \ \sum y = 149 + 339 + 357 + 445 + 380 = 1670 \ \sum xy = (2292 \cdot 149) + (7315 \cdot 339) + (11518 \cdot 357) + (13977 \cdot 445) + (19877 \cdot 380) = 826878 + 2486605 + 4112596 + 6205655 + 7552860 = 18686694 \ \sum x^2 = 2292^2 + 7315^2 + 11518^2 + 13977^2 + 19877^2 = 5240064 + 53462225 + 132490324 + 195496729 + 394160129 = 777913771 \]
Step 2: Compute the slope \( m \) and intercept \( b \)
Using the formulas: \[ m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} \] \[ b = \frac{\sum y - m \sum x}{n} \]
We can now plug in our calculated sums.
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Compute \( m \): \[ m = \frac{5 \cdot 18686694 - 43679 \cdot 1670}{5 \cdot 777913771 - (43679)^2} \] \[ m = \frac{93433470 - 72816630}{3889568855 - 1913400641} \] \[ m = \frac{20616840}{1976168214} \approx 0.0104 \]
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Compute \( b \): \[ b = \frac{1670 - 0.0104 \cdot 43679}{5} \] \[ b = \frac{1670 - 454.43}{5} \] \[ b = \frac{1215.57}{5} \approx 243.114 \]
Final Equation
Thus, the least squares regression line (rounded to the nearest thousandth) is: \[ y = 0.010x + 243.114 \]
So, the final answer is: \[ y = 0.010x + 243.114 \]