To determine the quadratic regression equation for the given data set, we'll find the best-fitting quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex] that describes the relationship between the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values. Given the detailed steps and calculations for this problem, let's analyze the provided data and coefficients to form our regression equation.
The given data points are:
- [tex]\( x \)[/tex] values: [2, 4, 6, 6, 7, 9, 8, 9, 11, 12]
- [tex]\( y \)[/tex] values: [32, 54, 85, 90, 95, 100, 90, 63, 29, 8]
The coefficients for the quadratic regression have been determined as follows:
- [tex]\( a \approx -3.01612 \)[/tex]
- [tex]\( b \approx 40.6817 \)[/tex]
- [tex]\( c \approx -45.8254 \)[/tex]
Using these coefficients, we can write the quadratic regression equation as:
[tex]\[ y = -3.01612x^2 + 40.6817x - 45.8254 \][/tex]
This equation best fits the pattern of the data provided.
Therefore, the correct quadratic regression equation for the data set is:
[tex]\[ y = -3.01612x^2 + 40.6817x - 45.8254 \][/tex]
The correct choice among the given options is:
[tex]\[ y = -3.01612x^2 + 40.6817x - 45.8254 \][/tex]
