To determine the equation of the line of best fit for the given data, we can use the method of least squares to find the best-fitting linear regression line. The given data points are:
- \( y \): 5, 9, 10, 10.5, 11
- \( x \): 1, 2, 3, 4, 5
Step 1: Calculate the necessary sums
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Calculate \( n \) (the number of data points): \[ n = 5 \]
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Calculate the sums needed: \[ \sum x = 1 + 2 + 3 + 4 + 5 = 15 \] \[ \sum y = 5 + 9 + 10 + 10.5 + 11 = 45.5 \] \[ \sum xy = (1 \times 5) + (2 \times 9) + (3 \times 10) + (4 \times 10.5) + (5 \times 11) = 5 + 18 + 30 + 42 + 55 = 150 \] \[ \sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55 \]
Step 2: Determine the slope (\( m \)) and y-intercept (\( b \))
The formulas for the slope \( m \) and y-intercept \( b \) of the line of best fit are given by:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ b = \frac{(\sum y) - m(\sum x)}{n} \]
Substituting the calculated sums:
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Calculate the slope \( m \): \[ m = \frac{5(150) - (15)(45.5)}{5(55) - (15)^2} \]
\[ = \frac{750 - 682.5}{275 - 225} = \frac{67.5}{50} = 1.35 \]
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Calculate the y-intercept \( b \): \[ b = \frac{45.5 - (1.35)(15)}{5} = \frac{45.5 - 20.25}{5} = \frac{25.25}{5} = 5.05 \]
Final Equation
Putting it all together, the equation of the line of best fit is:
\[ y = 1.35x + 5.05 \]
So, the final equation of the line of best fit is: \[ y = 1.35x + 5.05 \]
No rounding is needed, as values are already expressed to two decimal places.