Question
Find the equation of the line of best fit for the following data.
x y
3.5 5.85
6.9 1.25
2.1 7.41
0.6 9.73
5.3 3.01
8.2 0.09
4.7 4.62
(1 point)
Responses
y equals negative 1.2829 x plus 10.3021
Image with alt text: y equals negative 1.2829 x plus 10.3021
y equals negative 0.773033 x plus 8.00088
Image with alt text: y equals negative 0.773033 x plus 8.00088
y equals 1.28144 x minus 1.16414
Image with alt text: y equals 1.28144 x minus 1.16414
y equals negative 0.252953 x plus 5.69677
x y
3.5 5.85
6.9 1.25
2.1 7.41
0.6 9.73
5.3 3.01
8.2 0.09
4.7 4.62
(1 point)
Responses
y equals negative 1.2829 x plus 10.3021
Image with alt text: y equals negative 1.2829 x plus 10.3021
y equals negative 0.773033 x plus 8.00088
Image with alt text: y equals negative 0.773033 x plus 8.00088
y equals 1.28144 x minus 1.16414
Image with alt text: y equals 1.28144 x minus 1.16414
y equals negative 0.252953 x plus 5.69677
Answers
GPT-4o mini
To find the equation of the line of best fit for the given data, we can use the method of least squares to calculate the slope (m) and y-intercept (b) of the regression line \( y = mx + b \).
The formulas to calculate \( m \) and \( b \) are as follows:
1. The slope \( m \) is given by:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
2. The y-intercept \( b \) is given by:
\[
b = \frac{(\sum y) - m(\sum x)}{n}
\]
Where:
- \( n \) = number of data points
- \( \sum xy \) = sum of the product of each x and y
- \( \sum x \) = sum of all x values
- \( \sum y \) = sum of all y values
- \( \sum x^2 \) = sum of the squares of x values
Given your data:
```
x y
3.5 5.85
6.9 1.25
2.1 7.41
0.6 9.73
5.3 3.01
8.2 0.09
4.7 4.62
```
Let's calculate the necessary sums:
1. \( n = 7 \) (number of points)
2. \( \sum x = 3.5 + 6.9 + 2.1 + 0.6 + 5.3 + 8.2 + 4.7 = 31.3 \)
3. \( \sum y = 5.85 + 1.25 + 7.41 + 9.73 + 3.01 + 0.09 + 4.62 = 31.96 \)
4. \( \sum xy = (3.5*5.85) + (6.9*1.25) + (2.1*7.41) + (0.6*9.73) + (5.3*3.01) + (8.2*0.09) + (4.7*4.62) \)
- \( = 20.4975 + 8.625 + 15.561 + 5.838 + 15.953 + 0.738 + 21.7024 = 88.9439 \)
5. \( \sum x^2 = (3.5^2) + (6.9^2) + (2.1^2) + (0.6^2) + (5.3^2) + (8.2^2) + (4.7^2) \)
- \( = 12.25 + 47.61 + 4.41 + 0.36 + 28.09 + 67.24 + 22.09 = 182.01 \)
Now, we can substitute these sums into the formulas for \( m \) and \( b \):
Calculating \( m \):
\[
m = \frac{7(88.9439) - (31.3)(31.96)}{7(182.01) - (31.3)^2}
\]
\[
= \frac{622.6073 - 1000.088}{1274.07 - 979.69}
\]
\[
= \frac{-377.4807}{294.38} \approx -1.28144
\]
Calculating \( b \):
\[
b = \frac{31.96 - (-1.28144)(31.3)}{7}
\]
\[
= \frac{31.96 + 40.1166292}{7} \approx 10.3021
\]
Thus, the equation of the line of best fit is:
\[
y \approx -1.28144x + 10.3021
\]
Looking at the responses given, the correct one is:
**y equals negative 1.28144 x plus 10.3021**
The formulas to calculate \( m \) and \( b \) are as follows:
1. The slope \( m \) is given by:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
2. The y-intercept \( b \) is given by:
\[
b = \frac{(\sum y) - m(\sum x)}{n}
\]
Where:
- \( n \) = number of data points
- \( \sum xy \) = sum of the product of each x and y
- \( \sum x \) = sum of all x values
- \( \sum y \) = sum of all y values
- \( \sum x^2 \) = sum of the squares of x values
Given your data:
```
x y
3.5 5.85
6.9 1.25
2.1 7.41
0.6 9.73
5.3 3.01
8.2 0.09
4.7 4.62
```
Let's calculate the necessary sums:
1. \( n = 7 \) (number of points)
2. \( \sum x = 3.5 + 6.9 + 2.1 + 0.6 + 5.3 + 8.2 + 4.7 = 31.3 \)
3. \( \sum y = 5.85 + 1.25 + 7.41 + 9.73 + 3.01 + 0.09 + 4.62 = 31.96 \)
4. \( \sum xy = (3.5*5.85) + (6.9*1.25) + (2.1*7.41) + (0.6*9.73) + (5.3*3.01) + (8.2*0.09) + (4.7*4.62) \)
- \( = 20.4975 + 8.625 + 15.561 + 5.838 + 15.953 + 0.738 + 21.7024 = 88.9439 \)
5. \( \sum x^2 = (3.5^2) + (6.9^2) + (2.1^2) + (0.6^2) + (5.3^2) + (8.2^2) + (4.7^2) \)
- \( = 12.25 + 47.61 + 4.41 + 0.36 + 28.09 + 67.24 + 22.09 = 182.01 \)
Now, we can substitute these sums into the formulas for \( m \) and \( b \):
Calculating \( m \):
\[
m = \frac{7(88.9439) - (31.3)(31.96)}{7(182.01) - (31.3)^2}
\]
\[
= \frac{622.6073 - 1000.088}{1274.07 - 979.69}
\]
\[
= \frac{-377.4807}{294.38} \approx -1.28144
\]
Calculating \( b \):
\[
b = \frac{31.96 - (-1.28144)(31.3)}{7}
\]
\[
= \frac{31.96 + 40.1166292}{7} \approx 10.3021
\]
Thus, the equation of the line of best fit is:
\[
y \approx -1.28144x + 10.3021
\]
Looking at the responses given, the correct one is:
**y equals negative 1.28144 x plus 10.3021**