Simple Linear Regression

In mathematics, linear regression is the important topics in statistics. The process of determining the relationship between two variables is called as linear 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 simple linear regression tutor. Tutor will help the students with step by step solutions for the given problem. The following are the examples involved in simple linear regression tutor.

Simple linear regression tutor – Formula for Regression:

Here we help to know the regression formulas are

Formula for regression:

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

Where
x and y are the variables.
b = the slope of the regression line is also 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.

Simple linear regression tutor – Example problem:

Here we help to solve the example problem in simple linear regression tutor

Example 1:

To determine the regression equation by using the regression slope coefficient and intercept value.

X Values Y Values
22 14
32 15
42 16
52 17
72 19

For the given data set of data, solve the regression slope and intercept values.

Solution:

Let us count the number of values.
N = 5

Determine the values for XY, X

X Value Y Value   X*Y   X*X
  22   14     308    484
  32   15     480  1024
  42   16     672  1764
  52   17     884  2704
  72   19   1386  5184

Determine the following values `sumX` , `sumY` , `sumXY` , `sumX^2` .
`sumX` = 210
`sumY` = 81
`sumXY` = 3730
`sumX^2` = 11160

Substitute values in the slope formula
Slope (b) = `(NsumXY – (sumX) (sumY)) / (NsumX^2 – (sumX)^2)`
= `((5)*(3730)-(210)*(81))/((5)*(11160)-(210)^2)`
= `(18650 – 17010)/ (55800 – 44100)`
= `1640/11700`
b = 0.14
Substitute the values in the intercept formula given.
Intercept (a) = `(sumY – b (sumX)) / N `
= `(81 – 0.14 (210))/5`
= `(81 – 29.4)/5`
= `51.6/5`
a = 10.32

Substitute the Regression coefficient value and intercept value in the regression equation
Regression Equation(y) = a + bx
= 10.32 + 0.14x

Answer:

Slope (or) Regression coefficient (b) = 0.14

Intercept (a) = 10.32

Regression equation y = 10.32 + 0.14x