Certainly! Let's calculate the equation of the best fit line for the given data points [tex]\((x, y)\)[/tex].
Given:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline x & 3 & 6 & 9 & 12 & 15 & 18 \\
\hline y & -41 & -22 & -4 & 14 & 32 & 51 \\
\hline
\end{array}
\][/tex]
We want to find the equation of the line [tex]\(y = mx + b\)[/tex].
1. Calculate the necessary sums:
- Sum of [tex]\(x\)[/tex]-values:
[tex]\[
\sum{x} = 3 + 6 + 9 + 12 + 15 + 18 = 63
\][/tex]
- Sum of [tex]\(y\)[/tex]-values:
[tex]\[
\sum{y} = -41 + (-22) + (-4) + 14 + 32 + 51 = 30
\][/tex]
- Sum of product of corresponding [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
[tex]\[
\sum{xy} = (3 \cdot -41) + (6 \cdot -22) + (9 \cdot -4) + (12 \cdot 14) + (15 \cdot 32) + (18 \cdot 51) = 1275
\][/tex]
- Sum of [tex]\(x\)[/tex]-values squared:
[tex]\[
\sum{x^2} = 3^2 + 6^2 + 9^2 + 12^2 + 15^2 + 18^2 = 819
\][/tex]
Given [tex]\(n = 6\)[/tex] data points.
2. Calculate the slope [tex]\(m\)[/tex] of the best fit line:
The formula for the slope [tex]\(m\)[/tex] is:
[tex]\[
m = \frac{n \sum{xy} - (\sum{x})(\sum{y})}{n \sum{x^2} - (\sum{x})^2}
\][/tex]
Substitute the calculated values:
[tex]\[
m = \frac{6 \cdot 1275 - 63 \cdot 30}{6 \cdot 819 - 63^2}
\][/tex]
- Numerator:
[tex]\[
6 \cdot 1275 - 63 \cdot 30 = 7650 - 1890 = 5760
\][/tex]
- Denominator:
[tex]\[
6 \cdot 819 - 63^2 = 4914 - 3969 = 945
\][/tex]
[tex]\[
m = \frac{5760}{945} = 6.095238095238095
\][/tex]
3. Calculate the y-intercept [tex]\(b\)[/tex] of the best fit line:
The formula for the y-intercept [tex]\(b\)[/tex] is:
[tex]\[
b = \frac{\sum{y} - m \sum{x}}{n}
\][/tex]
Substitute the values:
[tex]\[
b = \frac{30 - (6.095238095238095 \cdot 63)}{6}
\][/tex]
- Calculate [tex]\(m \sum{x}\)[/tex]:
[tex]\[
m \sum{x} = 6.095238095238095 \cdot 63 = 384
\][/tex]
- Calculate [tex]\( \sum{y} - m \sum{x} \)[/tex]:
[tex]\[
30 - 384 = -354
\][/tex]
[tex]\[
b = \frac{-354}{6} = -59
\][/tex]
4. Final equation of the best fit line:
Therefore, the equation of the best fit line for the given data is:
[tex]\[
y = 6.095238095238095x - 59
\][/tex]
This step-by-step solution provides the detailed calculations leading to the slope [tex]\(m = 6.095238095238095\)[/tex] and the y-intercept [tex]\(b = -59\)[/tex].
