The Regression Line of y in x

To find the regression line of y in x, we need to solve the two normal equations obtained from the data:

1) 5a + 10b = 40
2) 10a + 25b = 95

We can solve these equations using various methods such as substitution, elimination, or matrix operations. Let's use the elimination method to find the values of a and b.

Solving the Normal Equations:

To eliminate the variable 'a', we can multiply the first equation by 2 and subtract it from the second equation:

2 * (5a + 10b) = 2 * 40
10a + 20b = 80

(10a + 25b) - (10a + 20b) = 95 - 80
5b = 15
b = 15 / 5
b = 3

Now, substitute the value of b back into the first equation:

5a + 10(3) = 40
5a + 30 = 40
5a = 40 - 30
5a = 10
a = 10 / 5
a = 2

The Regression Line:

Now that we have the values of a and b, we can form the equation of the regression line. The regression line represents the best-fit line that minimizes the sum of squared distances between the observed data points and the predicted values.

The equation of the regression line is given by:

y = a + bx

Substituting the values of a and b:

y = 2 + 3x

This equation represents the regression line of y in x. It indicates that for every unit increase in x, the predicted value of y increases by 3 units. The coefficient '2' represents the y-intercept, which is the predicted value of y when x is zero.

Summary:

- The regression line of y in x is given by the equation y = 2 + 3x.
- The values of a and b are obtained by solving the normal equations.
- The regression line represents the best-fit line that minimizes the sum of squared distances between the observed data points and the predicted values.
- The coefficient '2' represents the y-intercept, and '3' represents the slope of the line.