Let's analyze the given data set of hourly temperature measurements and find the quadratic regression equation that best fits these points.
The data points are:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
y & 20 & 16 & 10 & 0 & -7 & -20 \\
\hline
\end{array}
\][/tex]
In determining the quadratic regression equation, we need to find the coefficients \(a\), \(b\), and \(c\) of the general quadratic equation:
[tex]\[
y = ax^2 + bx + c
\][/tex]
The coefficients obtained are approximately:
1. \(a \approx -0.875\)
2. \(b \approx -3.596 \)
3. \(c \approx 20.179\)
Thus, the quadratic regression equation that fits the given data set is:
[tex]\[
y = -0.875 x^2 - 3.596 x + 20.179
\][/tex]
Comparing this equation with the options provided:
a. \(y = 0.795 x^2 + 3.796 x + 20.180\)
b. \(y = -0.795 x^2 - 3.760 x + 20.180\)
c. \(y = -0.875 x^2 - 3.596 x + 20.179\)
d. \(y = -0.795 x^2 - 3.796 x + 20.180\)
The correct answer is:
c. \(y = -0.875 x^2 - 3.596 x + 20.179\)
So, the quadratic regression equation that best represents the data set is:
[tex]\[
\boxed{y = -0.875 x^2 - 3.596 x + 20.179}
\][/tex]
