To find the predicted rainfall values using the given quadratic regression equation [tex]\(y = -0.77x^2 + 6.06x - 5.9\)[/tex], we will substitute each value of [tex]\(x\)[/tex] (representing the month) into the equation and solve for [tex]\(y\)[/tex] (the predicted rainfall).
1. Month 1 ([tex]\(x = 1\)[/tex]):
[tex]\[
y = -0.77(1)^2 + 6.06(1) - 5.9
\][/tex]
[tex]\[
y = -0.77 + 6.06 - 5.9
\][/tex]
[tex]\[
y \approx -0.61
\][/tex]
2. Month 2 ([tex]\(x = 2\)[/tex]):
[tex]\[
y = -0.77(2)^2 + 6.06(2) - 5.9
\][/tex]
[tex]\[
y = -0.77(4) + 12.12 - 5.9
\][/tex]
[tex]\[
y = -3.08 + 12.12 - 5.9
\][/tex]
[tex]\[
y \approx 3.14
\][/tex]
3. Month 3 ([tex]\(x = 3\)[/tex]):
[tex]\[
y = -0.77(3)^2 + 6.06(3) - 5.9
\][/tex]
[tex]\[
y = -0.77(9) + 18.18 - 5.9
\][/tex]
[tex]\[
y = -6.93 + 18.18 - 5.9
\][/tex]
[tex]\[
y \approx 5.35
\][/tex]
4. Month 4 ([tex]\(x = 4\)[/tex]):
[tex]\[
y = -0.77(4)^2 + 6.06(4) - 5.9
\][/tex]
[tex]\[
y = -0.77(16) + 24.24 - 5.9
\][/tex]
[tex]\[
y = -12.32 + 24.24 - 5.9
\][/tex]
[tex]\[
y \approx 6.02
\][/tex]
5. Month 5 ([tex]\(x = 5\)[/tex]):
[tex]\[
y = -0.77(5)^2 + 6.06(5) - 5.9
\][/tex]
[tex]\[
y = -0.77(25) + 30.3 - 5.9
\][/tex]
[tex]\[
y = -19.25 + 30.3 - 5.9
\][/tex]
[tex]\[
y \approx 5.15
\][/tex]
6. Month 6 ([tex]\(x = 6\)[/tex]):
[tex]\[
y = -0.77(6)^2 + 6.06(6) - 5.9
\][/tex]
[tex]\[
y = -0.77(36) + 36.36 - 5.9
\][/tex]
[tex]\[
y = -27.72 + 36.36 - 5.9
\][/tex]
[tex]\[
y \approx 2.74
\][/tex]
So, the predicted rainfall values for each month are:
- Month 1: [tex]\(-0.61 \, \text{cm}\)[/tex]
- Month 2: [tex]\(3.14 \, \text{cm}\)[/tex]
- Month 3: [tex]\(5.35 \, \text{cm}\)[/tex]
- Month 4: [tex]\(6.02 \, \text{cm}\)[/tex]
- Month 5: [tex]\(5.15 \, \text{cm}\)[/tex]
- Month 6: [tex]\(2.74 \, \text{cm}\)[/tex]
These values give an approximation of the rainfall values obtained using the quadratic regression model.
