Question

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

y
1
2
3
4
5
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Part 1
What is the linear function of the​ data?
yequals
  
enter your response herexplus​(
  
enter your response here​)
​(Type integers or​ decimals.)

Answer 1

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

Given the data points:

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

**Step 1: Calculate the means of \( x \) and \( y \)**

\[
\bar{x} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5
\]

\[
\bar{y} = \frac{1 + 2 + 3 + 4 + 5}{5} = \frac{15}{5} = 3
\]

**Step 2: Calculate the slope (m)**

The formula for the slope \( m \) is:

\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]

Next, we calculate each component:

\[
\begin{align*}
\sum (x_i - \bar{x})(y_i - \bar{y}) & = (1 - 5)(1 - 3) + (3 - 5)(2 - 3) + (5 - 5)(3 - 3) + (7 - 5)(4 - 3) + (9 - 5)(5 - 3) \\
& = (-4)(-2) + (-2)(-1) + (0)(0) + (2)(1) + (4)(2) \\
& = 8 + 2 + 0 + 2 + 8 = 20
\end{align*}
\]

\[
\sum (x_i - \bar{x})^2 = (1 - 5)^2 + (3 - 5)^2 + (5 - 5)^2 + (7 - 5)^2 + (9 - 5)^2 \\
= 16 + 4 + 0 + 4 + 16 = 40
\]

Now we can calculate \( m \):

\[
m = \frac{20}{40} = 0.5
\]

**Step 3: Calculate the y-intercept (b)**

The formula for the y-intercept \( b \) is:

\[
b = \bar{y} - m\bar{x}
\]

Substituting the values we have:

\[
b = 3 - 0.5 \cdot 5 = 3 - 2.5 = 0.5
\]

Finally, the equation of the linear function is:

\[
y = 0.5x + 0.5
\]

**Part 1 Answer:**

The linear function that best fits the data is:

\[
y = 0.5x + 0.5
\]