Answer :
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].
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].
