To find the line of best fit (regression line) for the given data, we can use linear regression. The standard form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Given the data points:
\[ \begin{align*} (2, 66) \ (4, 105) \ (4, 99) \ (4, 120) \ (5, 100) \ (5, 77) \ (5, 169) \ (6, 200) \ (6, 176) \ \end{align*} \]
Part A: Finding the Line of Best Fit
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Calculating the necessary sums:
- \(n\) (number of data points) = 9
- \(\sum x = 2 + 4 + 4 + 4 + 5 + 5 + 5 + 6 + 6 = 45\)
- \(\sum y = 66 + 105 + 99 + 120 + 100 + 77 + 169 + 200 + 176 = 1112\)
- \(\sum xy = 2 \times 66 + 4 \times 105 + 4 \times 99 + 4 \times 120 + 5 \times 100 + 5 \times 77 + 5 \times 169 + 6 \times 200 + 6 \times 176 = 132 + 420 + 396 + 480 + 500 + 385 + 845 + 1200 + 1056 = 4164\)
- \(\sum x^2 = 2^2 + 4^2 + 4^2 + 4^2 + 5^2 + 5^2 + 5^2 + 6^2 + 6^2 = 4 + 16 + 16 + 16 + 25 + 25 + 25 + 36 + 36 = 203\)
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Using the formulas for slope \(m\) and intercept \(b\): \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ b = \frac{(\sum y) - m(\sum x)}{n} \]
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Plugging in the values: \[ m = \frac{9(4164) - (45)(1112)}{9(203) - (45^2)} \] \[ m = \frac{37476 - 49940}{1827 - 2025} \] \[ m = \frac{-12464}{-198} \approx 62.92 \]
Now find \(b\): \[ b = \frac{1112 - 62.92 \cdot 45}{9} \] \[ b = \frac{1112 - 2831.4}{9} = \frac{-1719.4}{9} \approx -191.04 \]
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Thus, the line of best fit is: \[ y = 62.92x - 191.04 \]
Part B: Calculate the Residual for the Data Point (2, 66)
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Estimate Y-Value using the regression line: \[ Estimate = 62.92(2) - 191.04 = 125.84 - 191.04 \approx -65.20 \]
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Now calculate the residual: \[ \text{Residual Value} = \text{Actual Y-Value} - \text{Estimate Y-Value} \] \[ \text{Residual Value} = 66 - (-65.20) = 66 + 65.20 = 131.20 \]
Final Answers:
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Part A: The line of best fit is: \[ y = 62.92x - 191.04 \]
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Part B: The residual value for the point (2, 66) is: \[ \text{Residual Value} = 131.20 \]