Questions: Find the linear function that is the best fit for the data. What is the linear function of the data? y = []x + [] (Type integers or decimals.)

Solution Steps

To find the linear function that best fits the given data, we can use the method of least squares to determine the slope and intercept of the line. This involves calculating the means of the x and y values, the covariance of x and y, and the variance of x. Using these, we can derive the slope (m) and intercept (b) of the line in the form y = mx + b.

We need to find the linear function \( y = mx + b \) that best fits the given data points. This involves determining the slope \( m \) and the y-intercept \( b \) using the method of least squares.

The data points provided are:

• (9416, 0184)
• (774, 3184)
• (762, 7811)
• (539, 1097)

The formula for the slope \( m \) in the least squares method 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.

First, we calculate the necessary sums: \[ \sum x = 9416 + 774 + 762 + 539 = 11491 \] \[ \sum y = 0184 + 3184 + 7811 + 1097 = 12276 \] \[ \sum xy = (9416 \cdot 0184) + (774 \cdot 3184) + (762 \cdot 7811) + (539 \cdot 1097) \] \[ = 1732544 + 2466816 + 5950182 + 591283 = 10761825 \] \[ \sum x^2 = 9416^2 + 774^2 + 762^2 + 539^2 \] \[ = 88661136 + 599076 + 580644 + 290521 = 90111377 \]

Now, substituting these values into the formula for \( m \): \[ m = \frac{4(10761825) - (11491)(12276)}{4(90111377) - (11491)^2} \] \[ = \frac{43047300 - 141073596}{360445508 - 132091081} \] \[ = \frac{43047300 - 141073596}{228354427} \] \[ = \frac{-98026396}{228354427} \] \[ m \approx -0.4292 \]

The formula for the y-intercept \( b \) is: \[ b = \frac{\sum y - m(\sum x)}{n} \]

Substituting the values: \[ b = \frac{12276 - (-0.4292)(11491)}{4} \] \[ = \frac{12276 + 4933.0572}{4} \] \[ = \frac{17209.0572}{4} \] \[ b \approx 4302.2643 \]

Final Answer