To find the equation of the linear function that best fits the given data using linear regression, we can calculate the slope (m) and the y-intercept (b) using the formulas:
- Calculate the means of x and y.
- Use the following formulas for slope (m) and intercept (b):
\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
\[ b = \frac{\sum y - m(\sum x)}{N} \]
where:
- \( N \) is the number of data points,
- \( \sum xy \) is the sum of the products of each pair of x and y,
- \( \sum x \) is the sum of x values,
- \( \sum y \) is the sum of y values,
- \( \sum x^2 \) is the sum of the squares of x values.
Given the data:
- \( x: 1, 2, 3, 4, 5, 6 \)
- \( y: 80, 102, 129, 152, 186, 213 \)
First, compute the necessary sums:
\[ \sum x = 1 + 2 + 3 + 4 + 5 + 6 = 21 \] \[ \sum y = 80 + 102 + 129 + 152 + 186 + 213 = 862 \] \[ \sum xy = (1 \cdot 80) + (2 \cdot 102) + (3 \cdot 129) + (4 \cdot 152) + (5 \cdot 186) + (6 \cdot 213) = 80 + 204 + 387 + 608 + 930 + 1278 = 3487 \] \[ \sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91 \]
Now plug in the values into the formulas:
- \( N = 6 \)
- \( m = \frac{6(3487) - (21)(862)}{6(91) - (21)^2} \)
Calculating these:
\[ m = \frac{20922 - 18102}{546 - 441} = \frac{2810}{105} \approx 26.81 \]
Next, calculate \( b \):
\[ b = \frac{862 - 26.81(21)}{6} = \frac{862 - 562.01}{6} \approx \frac{299.99}{6} \approx 49.99 \]
Now we round \( m \) and \( b \) to two decimal places:
- \( m \approx 26.81 \)
- \( b \approx 50.00 \)
Finally, the equation of the linear function that best fits the data is:
\[ y = 26.81x + 50.00 \]
Thus, the final answer is:
y = 26.81x + 50.00