To create a linear model from the provided data points, we need to perform linear regression to find the equation of the line in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Given Data Points
- \( (4, 6) \)
- \( (6, 14) \)
- \( (8, 20) \)
- \( (10, 29) \)
- \( (12, 38) \)
- \( (14, 45) \)
Step 1: Calculate the Slope (m)
The formula for the slope \( m \) is: \[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
Where \( N \) is the number of data points.
Calculating necessary sums:
- \( N = 6 \)
- \( \sum x = 4 + 6 + 8 + 10 + 12 + 14 = 54 \)
- \( \sum y = 6 + 14 + 20 + 29 + 38 + 45 = 152 \)
- \( \sum xy = (46) + (614) + (820) + (1029) + (1238) + (1445) = 24 + 84 + 160 + 290 + 456 + 630 = 1644 \)
- \( \sum x^2 = 4^2 + 6^2 + 8^2 + 10^2 + 12^2 + 14^2 = 16 + 36 + 64 + 100 + 144 + 196 = 556 \)
Now substitute these values into the slope formula:
\[ m = \frac{6(1644) - (54)(152)}{6(556) - (54)^2} \]
Calculating this:
- \( m = \frac{9864 - 8208}{3336 - 2916} \)
- \( m = \frac{1656}{420} \)
- \( m \approx 3.943 \)
Step 2: Calculate the y-intercept (b)
The formula for the y-intercept \( b \) is: \[ b = \frac{\sum y - m(\sum x)}{N} \]
Substituting our values: \[ b = \frac{152 - 3.943(54)}{6} \] \[ b = \frac{152 - 213.762}{6} \] \[ b = \frac{-61.762}{6} \approx -10.294 \]
Final linear model:
Now we can write the linear model: \[ y \approx 3.943x - 10.294 \]
Rounded to three decimal places:
The final equation is: \[ y = 3.943x - 10.294 \]
Please enter the values as:
- \( y = 3.943x + (-10.294) \)
1 answer