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To determine the line of best fit for the given data points [tex]\((4, 85)\)[/tex], [tex]\((7, 92)\)[/tex], and [tex]\((14, 110)\)[/tex], we will calculate the slope and the intercept of the line that best matches these points.
Here's the step-by-step process to find the equation of the line of best fit:
1. Identify the Data Points:
- [tex]\( (x_1, y_1) = (4, 85) \)[/tex]
- [tex]\( (x_2, y_2) = (7, 92) \)[/tex]
- [tex]\( (x_3, y_3) = (14, 110) \)[/tex]
2. Calculate the Slope:
The formula for the slope [tex]\( m \)[/tex] of a line passing through [tex]\( (x_i, y_i) \)[/tex] points is:
[tex]\[ m = \frac{n(\sum{x_i y_i}) - (\sum{x_i})(\sum{y_i})}{n(\sum{x_i^2}) - (\sum{x_i})^2} \][/tex]
Where:
[tex]\[ n = 3 \][/tex] (since there are three points)
First, we calculate the required sums:
[tex]\[ \sum{x_i} = 4 + 7 + 14 = 25 \][/tex]
[tex]\[ \sum{y_i} = 85 + 92 + 110 = 287 \][/tex]
[tex]\[ \sum{x_i^2} = 4^2 + 7^2 + 14^2 = 16 + 49 + 196 = 261 \][/tex]
[tex]\[ \sum{x_i y_i} = 4 \times 85 + 7 \times 92 + 14 \times 110 = 340 + 644 + 1540 = 2524 \][/tex]
Now substitute these sums into the formula for the slope:
[tex]\[ m = \frac{3(2524) - (25)(287)}{3(261) - (25)^2} \][/tex]
[tex]\[ m = \frac{7572 - 7175}{783 - 625} \][/tex]
[tex]\[ m = \frac{397}{158} \approx 2.5126582278480996 \][/tex]
3. Calculate the Intercept:
The formula for the intercept [tex]\( b \)[/tex] is:
[tex]\[ b = \frac{\sum{y_i} - m(\sum{x_i})}{n} \][/tex]
Substitute the known values:
[tex]\[ b = \frac{287 - 2.5126582278480996 \times 25}{3} \][/tex]
[tex]\[ b = \frac{287 - 62.81645569620249}{3} \][/tex]
[tex]\[ b = \frac{224.18354430379751}{3} \approx 74.72784810126585 \][/tex]
4. Write the Equation of the Line of Best Fit:
[tex]\[ y = 2.5126582278480996x + 74.72784810126585 \][/tex]
Comparing the given options:
1. [tex]\( y = \frac{7}{3}x + 75.67 \approx 2.333x + 75.67 \)[/tex]
2. [tex]\( y = \frac{2}{3}x + 75.67 \approx 0.666x + 75.67 \)[/tex]
3. [tex]\( y = \frac{5}{3}x - 75.5 \approx 1.666x - 75.5 \)[/tex]
4. [tex]\( y = \frac{7}{4}x - 75.5 \approx 1.75x - 75.5 \)[/tex]
The equation that best matches the calculated equation of the line of best fit is:
[tex]\[ y = \frac{7}{3}x + 75.67 \approx 2.33x + 75.67 \][/tex]
Upon closer inspection, we find none of the provided options perfectly matches our found equation ([tex]\(2.5126582278480996x + 74.72784810126585\)[/tex]). Thus, the question might contain an error or the provided answer does not match any options exactly. Without the precise choice, the closest approximated result is the first one:
[tex]\[ y = \frac{7}{3}x + 75.67. \][/tex]
However, based on our exact calculations, here is the equation of the line of best fit:
[tex]\[ y = 2.5126582278480996x + 74.72784810126585. \][/tex]
Here's the step-by-step process to find the equation of the line of best fit:
1. Identify the Data Points:
- [tex]\( (x_1, y_1) = (4, 85) \)[/tex]
- [tex]\( (x_2, y_2) = (7, 92) \)[/tex]
- [tex]\( (x_3, y_3) = (14, 110) \)[/tex]
2. Calculate the Slope:
The formula for the slope [tex]\( m \)[/tex] of a line passing through [tex]\( (x_i, y_i) \)[/tex] points is:
[tex]\[ m = \frac{n(\sum{x_i y_i}) - (\sum{x_i})(\sum{y_i})}{n(\sum{x_i^2}) - (\sum{x_i})^2} \][/tex]
Where:
[tex]\[ n = 3 \][/tex] (since there are three points)
First, we calculate the required sums:
[tex]\[ \sum{x_i} = 4 + 7 + 14 = 25 \][/tex]
[tex]\[ \sum{y_i} = 85 + 92 + 110 = 287 \][/tex]
[tex]\[ \sum{x_i^2} = 4^2 + 7^2 + 14^2 = 16 + 49 + 196 = 261 \][/tex]
[tex]\[ \sum{x_i y_i} = 4 \times 85 + 7 \times 92 + 14 \times 110 = 340 + 644 + 1540 = 2524 \][/tex]
Now substitute these sums into the formula for the slope:
[tex]\[ m = \frac{3(2524) - (25)(287)}{3(261) - (25)^2} \][/tex]
[tex]\[ m = \frac{7572 - 7175}{783 - 625} \][/tex]
[tex]\[ m = \frac{397}{158} \approx 2.5126582278480996 \][/tex]
3. Calculate the Intercept:
The formula for the intercept [tex]\( b \)[/tex] is:
[tex]\[ b = \frac{\sum{y_i} - m(\sum{x_i})}{n} \][/tex]
Substitute the known values:
[tex]\[ b = \frac{287 - 2.5126582278480996 \times 25}{3} \][/tex]
[tex]\[ b = \frac{287 - 62.81645569620249}{3} \][/tex]
[tex]\[ b = \frac{224.18354430379751}{3} \approx 74.72784810126585 \][/tex]
4. Write the Equation of the Line of Best Fit:
[tex]\[ y = 2.5126582278480996x + 74.72784810126585 \][/tex]
Comparing the given options:
1. [tex]\( y = \frac{7}{3}x + 75.67 \approx 2.333x + 75.67 \)[/tex]
2. [tex]\( y = \frac{2}{3}x + 75.67 \approx 0.666x + 75.67 \)[/tex]
3. [tex]\( y = \frac{5}{3}x - 75.5 \approx 1.666x - 75.5 \)[/tex]
4. [tex]\( y = \frac{7}{4}x - 75.5 \approx 1.75x - 75.5 \)[/tex]
The equation that best matches the calculated equation of the line of best fit is:
[tex]\[ y = \frac{7}{3}x + 75.67 \approx 2.33x + 75.67 \][/tex]
Upon closer inspection, we find none of the provided options perfectly matches our found equation ([tex]\(2.5126582278480996x + 74.72784810126585\)[/tex]). Thus, the question might contain an error or the provided answer does not match any options exactly. Without the precise choice, the closest approximated result is the first one:
[tex]\[ y = \frac{7}{3}x + 75.67. \][/tex]
However, based on our exact calculations, here is the equation of the line of best fit:
[tex]\[ y = 2.5126582278480996x + 74.72784810126585. \][/tex]
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