Certainly, let's walk through the process of finding the best fit linear function for the given data step-by-step.
### Given Data Points:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 1 & 3 & 5 & 7 & 9 \\
\hline
y & 4 & 9 & 14 & 19 & 24 \\
\hline
\end{array}
\][/tex]
### Steps to Find the Linear Function:
1. Calculate the Mean of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
\bar{x} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5
\][/tex]
[tex]\[
\bar{y} = \frac{4 + 9 + 14 + 19 + 24}{5} = \frac{70}{5} = 14
\][/tex]
2. Compute the Slope (m) of the Line:
We use the formula for the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\][/tex]
Calculate the individual components:
[tex]\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = (1 - 5)(4 - 14) + (3 - 5)(9 - 14) + (5 - 5)(14 - 14) + (7 - 5)(19 - 14) + (9 - 5)(24 - 14)
\][/tex]
[tex]\[
= (-4)(-10) + (-2)(-5) + (0)(0) + (2)(5) + (4)(10)
\][/tex]
[tex]\[
= 40 + 10 + 0 + 10 + 40 = 100
\][/tex]
[tex]\[
\sum (x_i - \bar{x})^2 = (1 - 5)^2 + (3 - 5)^2 + (5 - 5)^2 + (7 - 5)^2 + (9 - 5)^2
\][/tex]
[tex]\[
= (-4)^2 + (-2)^2 + (0)^2 + (2)^2 + (4)^2
\][/tex]
[tex]\[
= 16 + 4 + 0 + 4 + 16 = 40
\][/tex]
So, the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{100}{40} = 2.5
\][/tex]
3. Compute the Y-intercept (b):
The y-intercept [tex]\( b \)[/tex] is found using the formula:
[tex]\[
b = \bar{y} - m \bar{x}
\][/tex]
Substitute the values we have:
[tex]\[
b = 14 - 2.5 \times 5
\][/tex]
[tex]\[
b = 14 - 12.5
\][/tex]
[tex]\[
b = 1.5
\][/tex]
### Linear Function of the Data:
Given the slope ( [tex]\( m = 2.5 \)[/tex] ) and the y-intercept ( [tex]\( b = 1.5 \)[/tex] ), the linear function that best fits the data is:
[tex]\[
y = 2.5x + 1.5
\][/tex]
So, the best fit linear function is:
[tex]\[
y = 2.5x + 1.5
\][/tex]
