Fitted Regression Equation

In mathematics, regression is one of the most interesting topics in statistics. The process of determining the relationship along with the two variables is called as regression. It is also one of the statistical analysis methods that can be used to assessing the association between the two different variables. In this article, we shall discuss about the fitted regression equation. It is used to help the students for the study about the fitted regression equation and the step by step explanations for the student doubts. The following are the example problem in fitted regression equation.
Fitted Regression Equation - Formula:

Formula for regression:

Regression Equation(y) = a + bx
Slope (b) = `(NsumXY - (sumX) (sumY)) / (NsumX^2 - (sumX)^2)`
Intercept(a) = `(sumY - b(sumX)) / N`

Looking out for more help on percentage equation in algebra by visiting listed websites.

Where
x and y are the variables.
b = the slope of the regression line is called as regression coefficient
a = intercept point of the regression line which is in the y-axis.
N = Number of values or elements
X = First Score
Y = Second Score
`sumXY` = Sum of the product of the first and Second Scores
`sumX` = Sum of First Scores
`sumY` = Sum of Second Scores
`sumX^2` = Sum of square First Scores.
Fitted Regression Equation - Example Problem:

Example:

Plot the fitted regression scatter plot for the given set of data and also solving the regression equation by finding the regression coefficients and slope values.
x y
32 22
35 26
37 29
49 34
56 40
89 53


Solution:

Let us count the number of values.
N = 6

Determine the values for xy, x2
x y xy x2
32 22 704 1024
35 26 910 1225
37 29 1073 1369
49 34 1666 2401
56 40 2240 3136
89 53 4717 7921


Find the following values `sumX` , `sumY` , `sumXY` , `sumX^2`.
`sumX` = 298
`sumY ` = 204
`sumXY` = 11310
`sumX^2 ` = 17076


Substitute values in the slope formula
Slope (b) = `(NsumXY - (sumX) (sumY)) / (NsumX^2 - (sumX)^2)`
= `((6)*(11310)-(298)*(204))/((6)*(17076)-(298)^2)`
= `(67860 - 60790)/(102456 - 88804)`
=  `7070/13652`
b = 0.5178
Substitute the values in the intercept formula given.
Intercept (a) = `(sumY - b (sumX)) / N`
= `(204 - 0.5178 (298))/6`
= `(204 - 154.3261)/6`
= `49.6738/6`
a = 8.2789

Substitute the Regression coefficient value and intercept value in the regression equation
Regression Equation(y) = a + bx
= 8.2789 + 0.5178x

For the given set of data, we can plot the graph

1. Take the X and Y values in the given set of data

2. In the graph, take the scale as for x and y axis as follows

3. Mark the x and y values in the graph

4. Plot the scatter plot and fitted regression line as in the diagram

Fitted regression equation - Example