Question

Find the linear function that is the best fit for the data.
x
1
3
5
7
9

y
2
3
4
5
6

Answer 1

To find the linear function \( y = mx + b \) that best fits the given data, we can use the least squares method to determine the slope \( m \) and the y-intercept \( b \).

### Step 1: Calculate the necessary sums

Given data:

\[
\begin{align*}
x: & \quad 1, 3, 5, 7, 9 \\
y: & \quad 2, 3, 4, 5, 6 \\
\end{align*}
\]

We calculate the following sums:

- \( n = 5 \) (number of points)
- Sum of \( x \): \( \sum x = 1 + 3 + 5 + 7 + 9 = 25 \)
- Sum of \( y \): \( \sum y = 2 + 3 + 4 + 5 + 6 = 20 \)
- Sum of \( xy \): \( \sum xy = (1)(2) + (3)(3) + (5)(4) + (7)(5) + (9)(6) = 2 + 9 + 20 + 35 + 54 = 120 \)
- 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: Calculate the slope \( m \) and y-intercept \( b \)

The formulas for \( m \) and \( b \) are as follows:

\[
m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]

\[
b = \frac{\sum y - m \sum x}{n}
\]

### Substituting the values:

1. Calculate \( m \):

\[
m = \frac{5(120) - (25)(20)}{5(165) - (25)^2} = \frac{600 - 500}{825 - 625} = \frac{100}{200} = 0.5
\]

2. Calculate \( b \):

\[
b = \frac{20 - 0.5(25)}{5} = \frac{20 - 12.5}{5} = \frac{7.5}{5} = 1.5
\]

### Final equation of the line

The equation of the best fit line is:

\[
y = 0.5x + 1.5
\]

Thus, the linear function that best fits the data is:

\[
\boxed{y = 0.5x + 1.5}
\]