Solved Examples on Regression Coefficients
1. Find the regression coefficients for the following data:
Age | Glucose Level |
|---|---|
|
25 |
90 |
|
30 |
65 |
|
35 |
75 |
|
40 |
79 |
|
45 |
81 |
|
50 |
87 |
Solution:
X (Age) | Y (Glucose Level) | XY | X2 | Y2 |
|---|---|---|---|---|
|
25 |
90 |
2250 |
625 |
8100 |
|
30 |
65 |
1950 |
900 |
4225 |
|
35 |
75 |
2625 |
1225 |
5625 |
|
40 |
79 |
3160 |
1600 |
6241 |
|
45 |
81 |
3645 |
2025 |
6561 |
|
50 |
87 |
4350 |
2500 |
7569 |
total = 225 | total = 477 | total = 17980 | total = 8875 | total = 38321 |
Using the formula above discussed, we find a (coefficient of X) and b (constant term) value:
a = 0.2114
b =71.57
The equation for the regression is :
- Y = a*X + b therefore,
- Y = 0.2114X + 71.57.
2. If the two regression coefficients between x and y are 0.6 and 0.4, then the coefficient of correlation between them is ?
Solution:
The two regression coefficients between x and y are 0.6 and 0.4
The correlation coefficient will be positive because both the coefficients are positive.
And the correlation coefficient is the geometric mean of both the coefficients. So the correlation coefficient is:
r = (0.6 * 0.4) 1/2
r = (0.24) 1/2
r = 0.489
3. From the given data, find the regression line
A | B |
|---|---|
|
6.25 |
4.03 |
|
6.5 |
4.02 |
|
6.5 |
4.02 |
|
6 |
4.04 |
|
6.25 |
4.03 |
|
6.25 |
4.03 |
Solution:
X (A) | Y (B) | XY | X2 | Y2 |
|---|---|---|---|---|
|
6.25 |
4.03 |
25.19 |
39.06 |
16.24 |
|
6.5 |
4.02 |
26.13 |
42.25 |
16.16 |
|
6.5 |
4.02 |
26.13 |
42.25 |
16.16 |
|
6 |
4.04 |
24.24 |
36 |
16.32 |
|
6.25 |
4.03 |
25.19 |
39.06 |
16.24 |
|
6.25 |
4.03 |
25.19 |
39.06 |
16.24 |
Total= 37.75 | Total= 24.17 | Total= 152.06 | Total= 237.69 | Total= 97.37 |
Using the formula above discussed, we find a (coefficient of X) and b (constant term) value:
a = -0.04.
b = 4.28
The equation for the regression is :
Y = a*X + b therefore,
Y = -0.04X + 4.28
Regression Coefficients
Regression coefficients in linear regression are the amounts by which variables in a regression equation are multiplied. Linear regression is the most commonly used form of regression analysis. Linear regression aims to determine the regression coefficients that result in the best-fitting line. These coefficients are helpful when estimating the value of an unknown variable using a known variable. This article explains regression coefficients and their formulas and provides related examples.
Table of Content
- What are Regression Coefficients?
- Regression Line
- Formula for Regression Coefficients
- Regression Coefficients Interpretation
- Steps to Calculate the Regression Coefficient
- Regression Coefficients in Different Types of Regression Models
- Solved Examples on Regression Coefficients