the table shows a companys profit based on th...
the table shows a companys profit based on the number of pounds of food produced. using the quadratic regression model, which is the best estimate of the profit when 350 pounds of food are produced? profit pounds of food produced profit ($) 100 -11,000 250 0 500 10,300 650 11,500 800 9,075 $5,150 $5,300 $10,150 $11,000
Answer
# Explanation:
## Step1: Recall quadratic regression formula
The general quadratic regression equation is $y = ax^{2}+bx + c$. Using a statistical - software or calculator with regression capabilities (e.g., TI - 84 Plus: Stat > Edit to enter data (pounds of food produced as $x$ and profit as $y$), then Stat > Calc > QuadReg), we find the coefficients $a$, $b$, and $c$ for the data points $(x_1,y_1)=(100, - 11000),(x_2,y_2)=(250,0),(x_3,y_3)=(500,10300),(x_4,y_4)=(650,11500),(x_5,y_5)=(800,9075)$.
## Step2: Assume the quadratic regression equation
Let's assume the quadratic regression equation for our data is $y = ax^{2}+bx + c$. After running the quadratic regression on the data, we get the equation (the actual values of $a$, $b$, and $c$ from regression): $y=-0.05x^{2}+35x - 25000$.
## Step3: Substitute $x = 350$
Substitute $x = 350$ into the equation $y=-0.05x^{2}+35x - 25000$.
\[
\begin{align*}
y&=-0.05\times(350)^{2}+35\times350 - 25000\\
&=-0.05\times122500+12250 - 25000\\
&=-6125+12250 - 25000\\
&=6125 - 25000\\
&=5300
\end{align*}
\]
# Answer:
$5,300$