To find the equation for the line of best fit and to estimate the median income for someone who is 32 years old, we can follow these steps.
A. Find the equation for the line of best fit
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Data Preparation:
Let's denote Age as \( x \) and Median Income as \( y \). From the given data, we have the following points:
\[ (26, 16.8), (27, 19.1), (28, 23.3), (29, 25.8), (30, 33.9) \]
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Calculating the necessary sums:
To find the best fit line, we will use the least squares method. The equation of the line will be in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Number of points, \( n = 5 \)
- Sum of \( x \): \[ \sum x = 26 + 27 + 28 + 29 + 30 = 140 \]
- Sum of \( y \): \[ \sum y = 16.8 + 19.1 + 23.3 + 25.8 + 33.9 = 118.9 \]
- Sum of \( xy \): \[ \sum xy = (26 \cdot 16.8) + (27 \cdot 19.1) + (28 \cdot 23.3) + (29 \cdot 25.8) + (30 \cdot 33.9) = 436.8 + 515.7 + 652.4 + 748.2 + 1017 = 3370.1 \]
- Sum of \( x^2 \): \[ \sum x^2 = 26^2 + 27^2 + 28^2 + 29^2 + 30^2 = 676 + 729 + 784 + 841 + 900 = 3930 \]
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Calculating the slope \( m \):
The formulas for the slope \( m \) and intercept \( b \) are: \[ m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} \] Plugging in the values: \[ m = \frac{5(3370.1) - (140)(118.9)}{5(3930) - (140)^2} \] \[ = \frac{16850.5 - 16646}{19650 - 19600} = \frac{204.5}{50} = 4.09 \]
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Calculating the y-intercept \( b \): \[ b = \frac{\sum y - m \sum x}{n} = \frac{118.9 - 4.09(140)}{5} \] \[ = \frac{118.9 - 572.6}{5} = \frac{-453.7}{5} = -90.74 \]
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Final equation: Thus, the equation of the line of best fit is: \[ y = 4.09x - 90.74 \]
B. Estimate the median income for someone 32 years old
To estimate the median income for someone who is 32 years old, we can plug \( x = 32 \) into the equation of the line we found.
\[ y = 4.09(32) - 90.74 \] \[ = 130.88 - 90.74 \] \[ = 40.14 \]
Conclusion
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The equation for the line of best fit is: \[ y = 4.09x - 90.74 \]
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The estimated median income for someone who is 32 years old is approximately: \[ $40,140 \] (Note: Ensure consistency with units as the median income might be represented in a certain range based on context or scaling).