To determine the quadratic regression equation for the given data set, we need to find the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] in the quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex].
Given are the data points:
[tex]\[ x = [1, 3, 2, 5, 4, 5, 6, 8, 8, 10] \][/tex]
[tex]\[ y = [-98, -103, -108, -129, -144, -173, -222, -280, 339, -420] \][/tex]
By fitting the quadratic regression model to the data, we determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. The resulting coefficients are:
[tex]\[ a \approx -4.96981 \][/tex]
[tex]\[ b \approx 43.07625 \][/tex]
[tex]\[ c \approx -186.83502 \][/tex]
Thus, the quadratic regression equation that best fits the data set is:
[tex]\[ y = -4.96981 x^2 + 43.07625 x - 186.83502 \][/tex]
Now comparing the provided equations to the one we calculated:
1. [tex]\( y = -3.19847 x^2 + 5.17384 x - 95.8074 \)[/tex]
2. [tex]\( y = -3.19847 x^2 + 4.72843 x - 95.8074 \)[/tex]
3. [tex]\( y = -3.85746 x^2 + 5.17384 x - 95.8074 \)[/tex]
4. [tex]\( y = -3.85746 x^2 + 4.72843 x - 95.8074 \)[/tex]
None of these match our determined equation exactly.
Therefore, it appears there might be a mismatch between the provided options and the actual fitting equation based on the data set provided. If you need to confirm, you may need to re-check the data fitting process or consult additional resources.
