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`

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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

Variable Pairs

Variable pairs determines the concept of an ordered pair and it shows you to plot ordered pairs  on a graph. The variable determine the value of the particular terms of the equation.

The variable pair is a set of two variable like (x, y) written in parentheses. The variable in terms of graph represent the first value as x and the second value as y. (2,3) represents x axis is 2 units long and y axis is 3 units long.

Example Problem for Variable Pairs:

Solve for the ordered pair from the following equations:

4x + 3y = 12

2x + 3y =  8 solve for x and y

Solution:

Given that 4x + 3y = 12--------------------- (1)

2x + 3y =   8--------------------- (2)

Subtract the first equation and second equation,



4x + 3y = 12

2x + 3y =   8

________________

2x  = 4

________________

That is

4x – 2x = 2x

3y – 3y = 0

12 – 8   = 4

Now we have the equation 2x = 4 after subtraction.

2x = 4

Divide by 2 on both sides

x = `4 / 2`

x = 2

Substitute the x = 2 in the first equation, we will get the value of y variable.

First equation 4x + 3y = 12

4(2) + 3y = 12

8 + 3y = 12

3y = 12 – 8

3y = 4

y = `4 / 3`

The solution is x = 2 and y = `4 /3` .

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Example Problem for Variable Pairs:

Solve for the ordered pair from the following equations:

8x + 2y = 4

x + 3y =   5

Solution:

Given that 8x + 2y = 4--------------------- (1)

x + 3y =   5--------------------- (2)

Subtract the first equation and second equation,

3 * (1)    24x + 6y = 12

2 * (2)      2x + 6y =-10

________________

22x   = 22

________________

That is

24x – 2x = 22x

6y – 6y = 0

12 – (-10)  = 22

Now we have the equation 22x = 22 after subtraction.

Divide by 22 on both sides. we get

x = 1

Substitute the x = 22 in the first equation,

First equation 8x + 3y = 4

8(22) + 3y = 4

176 + 3y = 4

3y = 4 – 176

3y = -172

y = `-172 / 3`

The solution is x = 1 and y = -`172 /3` .

The value of variable pairs (x,y) is (1,-`172/3` )

Linearly Independent Determinant

In mathematics, determinants are linearly independent if none of the determinants can be obtained from the others. When determinants are linearly independent, then each determinant contains new information about the variables. For example

A = [[0],[0],[2]] B = [[2],[-4],[2]] C = [[0],[4],[-4]] D = [[8],[4],[3]] .

Here, the determinants A, B and C are linearly independent but the determinant of D is not since determinant of D is equals to 9A + 4B + 5C. Following example and practice problems will help you to study about linearly independent determinant.
Example Problems of Linearly Independent Determinant:

Example problem 1:

Show that the determinant [[2,3],[4,5]] is linearly dependent or not using Wronskian determinant method.

Solution:

Step 1: Given determinant

[[2,3],[4,5]] .

Step 2: Condition of Wronskian determinant

If the value of the determinant is equal to 0, then the determinant is linearly dependent.

If the value of the determinant is not equal to 0, then the determinant is linearly independent.

Step 3: Calculate the value of the determinant.

[[2,3],[4,5]] = (2 * 5) - (3 * 4)

= -2

Since the value of the determinant is not equal to 0, the given determinant is linearly independent.

Step 4: Solution

Hence, the given determinant [[2,3],[4,5]] is linearly independent.

Example problem 2:

Show that the determinant [[1,4],[5,20]] is linearly dependent or not using Wronskian determinant method.

Solution:

Step 1: Given determinant

[[1,4],[5,20]] .

Step 2: Condition of Wronskian determinant

If the value of the determinant is equal to 0, then the determinant is linearly dependent.

If the value of the determinant is not equal to 0, then the determinant is linearly independent.

Step 3: Calculate the value of the determinant.

[[1,4],[5,20]] = (1 * 20) - (4 * 5)

= 0

Since the value of the determinant is equal to 0, the given determinant is linearly dependent.

Step 4: Solution

Hence, the given determinant [[1,4],[5,20]] is linearly dependent.
Practice Problems of Linearly Independent Determinant:

1) Show that the determinant [[4,3],[2,8]] is linearly dependent or not using Wronskian determinant method.

2) Show that the determinant [[2,5],[4,10]] is linearly dependent or not using Wronskian determinant method.

Solutions:

1) The given determinant [[4,3],[2,8]] is linearly independent.

2) The given determinant [[2,5],[4,10]] is linearly dependent.

Rules for Factoring Trinomials

Trinomial is nothing but if we are having the function with three terms we can say that is a trinomial. Here we are going to learn what the rules available in factoring the trianomial are. There are three rules available for factoring trinomials. We will see the rules for factoring the trinomials with appropriate examples. Factoring the trinomial is depends on the degree of polynomials.

Rules for Factoring Trinomials:

If we want to factor a trinomial we have to start from the highest exponents. For factoring the trinomial first we have to find the greatest common factor and we have to simplify.

Example:

Factor the following trinomial 6x8 + 30 x7 + 36 x6

Solution:

First we have to find the factors of the greatest value

6x8 = 2. 3.  x2. x6

30x7 = 2. 3. 5.  x. x6

36 x6 = 2. 3. 6. x6

And here GCF of the variables 6x8, 30 x7, 36 x6 is 6x6

So the factor value is 6x6 (x2 + 5x + 6) = 6x6 (x + 2) (x + 3)

Rule 2 for factoring trinomial:

In rule 2 we have to see the factoring rules in algebra based on grouping the value. Here we are going to group the given trinomial and we have to factor this.

Example:

Factor the trinomial 3y2 + y – 4

Solution:

3y2 + y – 4

This is 3y2 + 4y – 3y - 4

So we get 3y(y + 4) - (3y + 4)

From this y (3y - 1) (3y + 4)

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Other Rules for Factoring Trinomial:

Rule 3 for factoring trinomial:

If the given trinomial expression starts with the co – efficient of x2 we have to start from multiplication.

Example:

Factor the trinomial 10x2 - 14x + 4

Solution:

Given expression is 10x2 - 14x + 4

First we have to multiply 10 and 4

We get x2 - 13x + 40

Now if we divide this 10x2 - 4x – 10x + 4

x(10x –4) – (10x – 4)

(10x – 4) (x – 1)

These rules are helping us to factor the trinomial easily.

Measuring Angles Interactive

The interactive of measuring the exterior angles are the angles between any sides of shapes, and a line extended from the other side. Interactive figures exterior angles are any straight sides with flat shape. Polygon measuring exterior angle sum is 360°. In other words Polygon measuring of sum exterior angle is one full revolution.  The measuring angles are made with an extension of a side and its corresponding adjacent sides are the exterior angle.

Overview for Measuring Angles Interactive:

If you can measuring the interior and the exterior angle of straight line gives 180 degree. The Interactive regular polygons or objects exterior angles are the congruent, and polygon measuring exterior angle sum is 360°. So each polygon of measuring exterior angle is 360 divided by the n, the number of sides. Any figure interior angle is supplementary to an exterior angle at that vertex is 180.  Interior angles and exterior angles are always complementary.
Example Problems for Measuring Angles Interactive:

1) Measuring exterior angles of polygon, that the polygon angles are 80, x, 95, 125 degrees

Solution:

All polygons measuring of exterior angle sum is 360 degrees.

So add all the exterior angles and equate to the 360 degrees.

The given exterior angles are 80, x, 95, 125 degrees.

80+95+125+x = 360

300+x =x360

Subtracted both sides by 300

x = 360-300

x = 60`^@`

2) Measuring of exterior angles of polygon, that the polygon angles are 40, 2x, 3x, 2x, x degrees

Solution:

All polygon measuring exterior angle sums is 360 degrees.

So add up all the exterior angles and equate to the 360 degrees.

Given exterior angles are 40, 2x, 3x, 2x, x degrees.

40+x+2x+3x+2x = 360

40+8x = 360

Subtract both sides by 40 degrees

8x = 320

Divide both sides by 8

X = 40

Therefore each angle is

X = 40

2x = 2*40 = 80

3x = 3*40 =120

Each exterior angle is 40°, 40°, 80°, 120°, and 80°.


3) Measuring exterior angles of polygon, that the polygon angles are 80, x, 95, 125 degrees

Solution:

Given figure angle is four sided shape.

The sum of interior angle any four sided figure is 360 degree

All polygons measuring of interior angle sum is 360 degrees.

So add all the interior angles and equate to the 360 degrees.

The given interior angles are 80, x, 95, 115 degrees.

80+95+115+x  =360

290+x = 360

Subtracted both sides by 290

x = 360-290

x = 70`^@`