To find the linear function that best fits the given data points, we can use the least squares method to determine the slope (m) and y-intercept (b) of the linear equation in the form:
\[ y = mx + b \]
Given Data:
- x values: 1, 3, 5, 7, 9
- y values: 2, 5, 8, 11, 14
Step 1: Calculate the necessary sums
We need to calculate the following sums:
- \( N \): Number of data points
- \( \sum x \): Sum of x values
- \( \sum y \): Sum of y values
- \( \sum xy \): Sum of the product of x and y values
- \( \sum x^2 \): Sum of the squares of the x values
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Count of Data Points, N: \( N = 5 \)
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Sum of x values: \( \sum x = 1 + 3 + 5 + 7 + 9 = 25 \)
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Sum of y values: \( \sum y = 2 + 5 + 8 + 11 + 14 = 40 \)
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Sum of xy values: \( \sum xy = (1 \cdot 2) + (3 \cdot 5) + (5 \cdot 8) + (7 \cdot 11) + (9 \cdot 14) \) \[ = 2 + 15 + 40 + 77 + 126 = 260 \]
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Sum of x² values: \( \sum x^2 = (1^2) + (3^2) + (5^2) + (7^2) + (9^2) \) \[ = 1 + 9 + 25 + 49 + 81 = 165 \]
Step 2: Calculate the slope (m) and intercept (b)
Using the formulas for m and b:
\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
\[ b = \frac{(\sum y) - m(\sum x)}{N} \]
Calculating m:
\[ m = \frac{5(260) - 25(40)}{5(165) - (25)^2} \] \[ = \frac{1300 - 1000}{825 - 625} \] \[ = \frac{300}{200} = 1.5 \]
Calculating b:
\[ b = \frac{40 - 1.5(25)}{5} \] \[ = \frac{40 - 37.5}{5} \] \[ = \frac{2.5}{5} = 0.5 \]
Final Equation
So, the linear function that best fits the data is:
\[ y = 1.5x + 0.5 \]
Answer Format
Thus, in the required format:
y = \( 1.5x + (0.5) \)