Question

Consider the following x, y data pairs: x = [2, 5, 6] and y = [6, 4, 8]. Find the least-squares equation for these data. (Use 3 decimal places.)

Answer 1

The least-squares equation for the given x, y data pairs is ( y = 4.333x + 1.333 ).

Step-by-step explanation:

To find the least-squares equation, we use the formula ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept. Using the given x, y data pairs, we calculate the slope ( m ) and y-intercept ( b ).

The slope is found by the formula
`\( m = (n(\sum xy) - (\sum x)(\sum y))/(n(\sum x^2) - (\sum x)^2) \)`, and the y-intercept is then calculated using
`\( b = (\sum y - m(\sum x))/(n) \)`. Substituting the values, we find
`\( m \approx 4.333` and
`\( b \approx 1.333 \)`, resulting in the least-squares equation ( y = 4.333x + 1.333 ).

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables.

The least-squares method minimizes the sum of the squares of the residuals (the differences between observed and predicted values) to find the best-fitting line. This technique is commonly used in data analysis and is a fundamental tool in regression analysis.