To find the equation for the line of best fit based on the provided points, we first need to determine the slope (m) and y-intercept (b) of the line in the form \( y = mx + b \).
We have the following points:
- \( a = (1, 30) \)
- \( b = (2, 35) \)
- \( c = (5, 55) \)
- \( d = (7, 60) \)
- \( e = (10, 70) \)
Step 1: Calculate the slope (m)
The formula for the slope \( m \) is given by:
\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
where \( N \) is the number of points.
Calculate the necessary sums:
- Count (N): There are 5 points.
- Sum of x values: \( \sum x = 1 + 2 + 5 + 7 + 10 = 25 \)
- Sum of y values: \( \sum y = 30 + 35 + 55 + 60 + 70 = 250 \)
- Sum of x*y:
- \( \sum xy = (1)(30) + (2)(35) + (5)(55) + (7)(60) + (10)(70) \)
- \( = 30 + 70 + 275 + 420 + 700 = 1495 \)
- Sum of x^2:
- \( \sum x^2 = 1^2 + 2^2 + 5^2 + 7^2 + 10^2 \)
- \( = 1 + 4 + 25 + 49 + 100 = 179 \)
Step 2: Substitute into the slope formula:
\[ m = \frac{5(1495) - (25)(250)}{5(179) - (25)^2} \] \[ m = \frac{7475 - 6250}{895 - 625} \] \[ m = \frac{1225}{270} \approx 4.54 \]
Step 3: Calculate the y-intercept (b)
The y-intercept can be found using the formula:
\[ b = \frac{\sum y - m(\sum x)}{N} \]
Substituting the values we calculated:
\[ b = \frac{250 - 4.54(25)}{5} \] \[ b = \frac{250 - 113.5}{5} \] \[ b = \frac{136.5}{5} \approx 27.3 \]
Final equation for the line of best fit:
Combining slope and y-intercept gives:
\[ y \approx 4.54x + 27.3 \]
Thus, the equation for the line of best fit is:
\[ y = 4.54x + 27.3 \]