IDNLearn.com makes it easy to find answers and share knowledge with others. Explore thousands of verified answers from experts and find the solutions you need, no matter the topic.

Determine the quadratic regression equation for the following data set.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline [tex]$x$[/tex] & 3 & 5 & 6 & 5 & 8 & 7 & 10 & 11 & 11 & 12 \\
\hline [tex]$y$[/tex] & -126 & -150 & -172 & -229 & -273 & -335 & -420 & -506 & -598 & -708 \\
\hline
\end{tabular}

A. [tex] y = -6.36897 x^2 + 41.3118 x - 200.937 [/tex]

B. [tex] y = -6.36897 x^2 + 41.3118 x - 188.429 [/tex]

C. [tex] y = -6.36897 x^2 + 37.3389 x - 200.937 [/tex]

D. [tex] y = -6.36897 x^2 + 37.3389 x - 188.429 [/tex]


Sagot :

To determine the quadratic regression equation for the given data set, we follow these steps:

1. Define the form of the quadratic equation:
The general form of a quadratic equation is:
[tex]\[ y = ax^2 + bx + c \][/tex]

2. Fit the quadratic model to the data:
Using the given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values, we fit the quadratic model to determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].

3. Determine the coefficients:
The quadratic model, after fitting it to the given data points, yields the following coefficients:
[tex]\[ a = -6.36897 \][/tex]
[tex]\[ b = 37.3389 \][/tex]
[tex]\[ c = -200.937 \][/tex]

4. Substitute the coefficients back into the quadratic equation:
Substituting [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the general form of the quadratic equation, we get:
[tex]\[ y = -6.36897x^2 + 37.3389x - 200.937 \][/tex]

Therefore, the correct quadratic regression equation for the given data set is:
[tex]\[ y = -6.36897x^2 + 37.3389x - 200.937 \][/tex]

So, the correct answer is:
[tex]\[ \boxed{y = -6.36897x^2 + 37.3389x - 200.937} \][/tex]