To find the equation of the line of best fit for the given data points, we'll use the method of least squares to determine the slope ([tex]\(m\)[/tex]) and y-intercept ([tex]\(c\)[/tex]) of the best-fit line. Here are the steps:
1. List the given data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
5 & 4 \\
\hline
6 & 6 \\
\hline
9 & 9 \\
\hline
10 & 11 \\
\hline
14 & 12 \\
\hline
\end{array}
\][/tex]
2. Calculate the slope ([tex]\(m\)[/tex]) and y-intercept ([tex]\(c\)[/tex]):
The formula for the slope [tex]\(m\)[/tex] and y-intercept [tex]\(c\)[/tex] can be derived from the normal equations of linear regression:
[tex]\[
m = \frac{N\sum(xy) - \sum(x)\sum(y)}{N\sum(x^2) - (\sum(x))^2}
\][/tex]
[tex]\[
c = \frac{\sum(y) - m\sum(x)}{N}
\][/tex]
where [tex]\(N\)[/tex] is the number of data points.
3. Perform the calculations to obtain [tex]\(m\)[/tex] and [tex]\(c\)[/tex]:
By processing the given data, the calculations yield the following results:
- Slope ([tex]\(m\)[/tex]): 0.894 (rounded to three decimal places)
- Y-intercept ([tex]\(c\)[/tex]): 0.535 (rounded to three decimal places)
4. Form the equation of the line of best fit:
With [tex]\(m = 0.894\)[/tex] and [tex]\(c = 0.535\)[/tex], the equation of the line of best fit becomes:
[tex]\[
y = 0.894x + 0.535
\][/tex]
5. Identify the correct answer choice:
Given the options:
- A. [tex]\(y = 0.535x + 0.894\)[/tex]
- B. [tex]\(y = 0.894x + 0.535\)[/tex]
- C. [tex]\(y = -0.535x + 0.894\)[/tex]
- D. [tex]\(y = -0.894x + 0.535\)[/tex]
The correct answer is B. [tex]\(y = 0.894x + 0.535\)[/tex].
