To answer this question, we need to evaluate the quadratic regression equation given:
[tex]\[ y = -1.34x^2 + 10.75x - 11.3 \][/tex]
for [tex]\( x = 10 \)[/tex].
First, substitute [tex]\( x = 10 \)[/tex] into the equation:
[tex]\[
y = -1.34(10)^2 + 10.75(10) - 11.3
\][/tex]
Calculate [tex]\((10)^2\)[/tex]:
[tex]\[
(10)^2 = 100
\][/tex]
Then multiply this by [tex]\(-1.34\)[/tex]:
[tex]\[
-1.34 \times 100 = -134
\][/tex]
Now multiply [tex]\( 10.75 \times 10 \)[/tex]:
[tex]\[
10.75 \times 10 = 107.5
\][/tex]
Now, substitute these values back into the equation:
[tex]\[
y = -134 + 107.5 - 11.3
\][/tex]
Add these together:
[tex]\[
y = -134 + 107.5 - 11.3 = -37.8
\][/tex]
So, the predicted number of owls for year 10 using this model is approximately [tex]\( -37.8 \)[/tex].
Let's analyze this result: a population number cannot be negative because it doesn't make sense to have a negative count of living organisms. Therefore, even though the calculation gives us a result mathematically, a negative value for the population of owls is not feasible in a real-world scenario.
Given the options provided:
A. Yes, because the owl population is endangered.
B. Yes, because that is the result of substituting [tex]\( x=10 \)[/tex].
C. No, because the owls went to live somewhere else.
The correct answer is:
None of the given choices appropriately explain why a negative population makes sense. This model's result suggests that the population cannot be negative. Therefore, the correct statement would be: "No, because a population cannot be negative."
