Question

Hello! I got this answer but I am unsure as to how I got the answer.

Answer 1

Given a quadratic equation: y = ax² + bx + c, "a", "b" and "c" are the coefficients of the regression and can be estimated using the formulas below.

a = {[Σ x²y * Σ xx ] - [Σ xy * Σ xx² ] } / { [ Σ xx * Σ x²x²] - [Σ xx² ]²}

b = {[Σ xy * Σ x²x² ] - [Σ x²y * Σ xx² ] } / { [ Σ xx * Σ x²x²] - [Σ xx² ]²}

c = [Σ y / n ] - { b * [ Σ x / n ] } - { a * [ Σ x² / n ]}

Where:

Σ xx = [Σ x²] - [ ( Σ x )² / n]

Σ xy = [Σ x y] - [ ( Σ x * Σ y ) / n]

Σ xx² = [Σ x³] - [ ( Σ x² * Σ x ) / n]

Σ x²y = [Σ x²y] - [ ( Σ x² * Σ y ) / n]

Σ x²x² = [Σ x⁴] - [ ( Σ x² )² / n]

And:

x and y are the Variables.

n = Number of Values or Elements

Σ x = Sum of x Scores

Σ y = Sum of y Scores

Σ x² = Sum of Square of x

Σ x³ = Sum of Cube of x

Σ x⁴ = Sum of Power Four of x

Σ xy= Sum of the Product of x and y

Σ x²y = Sum of Square of x and y

To solve the problem, follow the steps below.

Step 01: Find n and the values of x and y.

n = 6.

x = 8, 10, 12, 14, 16, 18

y = 45.94, 56.84, 63.88, 67.07, 66.4, 61.87

Step 02: Find Σ x, Σ y, Σ x², Σx³, Σ x⁴, Σ xy, Σ x²y

* (,) represents a decimal point.

Step 03: Find Σ xx, Σ xy, Σ xx², Σ x²y, Σ x²x²

Substitute the values above in the equations for each variable:

Σ xx = [Σ x²] - [ ( Σ x )² / n]

Σ xy = [Σ x y] - [ ( Σ x * Σ y ) / n]

Σ xx² = [Σ x³] - [ ( Σ x² * Σ x ) / n]

Σ x²y = [Σ x²y] - [ ( Σ x² * Σ y ) / n]

Σ x²x² = [Σ x⁴] - [ ( Σ x² )² / n]

$\begin{gathered} \sum_^xx=\operatorname{\sum}_^x^2-\frac{(\operatorname{\sum}_^x)^2}{n} \\ \sum_^xx=1084-\frac{78^2}{6} \\ \sum_^xx=1084-\frac{6084}{6} \\ \sum_^xx=1084-1014 \\ \sum_^xx=70 \end{gathered}$

Doing the same for the other variables, you have the results:

* (,) represents a decimal point.

Step 03: Find "a".

a = {[Σ x²y * Σ xx ] - [Σ xy * Σ xx² ] } / { [ Σ xx * Σ x²x²] - [Σ xx² ]²}

$\begin{gathered} a=\frac{2611.55*70-111.52*1820}{70*47917.3-1820^2} \\ a=-0.482 \end{gathered}$

Step 04: Find "b".

b = {[Σ xy * Σ x²x² ] - [Σ x²y * Σ xx² ] } / { [ Σ xx * Σ x²x²] - [Σ xx² ]²}

$\begin{gathered} b=\frac{111.52*47917.3-2611.55*1820}{70*47917.3-1820^2} \\ b=14.125 \end{gathered}$

Step 05: Find "c".

c = [Σ y / n ] - { b * [ Σ x / n ] } - { a * [ Σ x² / n ]}

$\begin{gathered} c=\frac{362}{6}-14.125*\frac{78}{6}-(-0.482)*\frac{1084}{6} \\ c=-36.209 \end{gathered}$

Answer:

The equation is:

$-0.482x^2+14.125x-36.209$