To find the linear function that best fits the data, we can use the method of least squares to find the slope (m) and y-intercept (b) of the line in the form \( y = mx + b \).
Given Data Points:
- \( (1, 3) \)
- \( (3, 4) \)
- \( (5, 5) \)
- \( (7, 6) \)
- \( (9, 7) \)
Step 1: Calculate the necessary sums
Let \( n \) be the number of data points, and calculate the following:
- \( n = 5 \) (the number of points)
- Sum of \( x \): \[ \sum x = 1 + 3 + 5 + 7 + 9 = 25 \]
- Sum of \( y \): \[ \sum y = 3 + 4 + 5 + 6 + 7 = 25 \]
- Sum of \( xy \): \[ \sum xy = (1)(3) + (3)(4) + (5)(5) + (7)(6) + (9)(7) = 3 + 12 + 25 + 42 + 63 = 145 \]
- Sum of \( x^2 \): \[ \sum x^2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165 \]
Step 2: Apply the formulas for slope (m) and intercept (b)
The formulas for the slope \( m \) and y-intercept \( b \) are given by:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]
\[ b = \frac{(\sum y) - m(\sum x)}{n} \]
Step 3: Calculate the slope (m)
Plugging in the values:
\[ m = \frac{5(145) - (25)(25)}{5(165) - (25)^2} \] \[ m = \frac{725 - 625}{825 - 625} \] \[ m = \frac{100}{200} = 0.5 \]
Step 4: Calculate the intercept (b)
Now that we have \( m \):
\[ b = \frac{25 - 0.5(25)}{5} \] \[ b = \frac{25 - 12.5}{5} = \frac{12.5}{5} = 2.5 \]
Final Equation
Thus, the linear function that best fits the data is:
\[ y = 0.5x + 2.5 \]
Thus, your answer is: \[ y = 0.5x + 2.5 \]