the table shows a companys profit based on the number of pounds of food produced.\nprofit\npounds of food…

the table shows a companys profit based on the number of pounds of food produced.\nprofit\npounds of food produced\tprofit ($)\n100\t-11,000\n250\t0\n500\t10,300\n650\t11,500\n800\t9,075\nusing the quadratic regression model, which is the best estimate of the profit when 350 pounds of food are produced?\n$5,150\n$5,300\n$10,150\n$11,000

the table shows a companys profit based on the number of pounds of food produced.\nprofit\npounds of food produced\tprofit ($)\n100\t-11,000\n250\t0\n500\t10,300\n650\t11,500\n800\t9,075\nusing the quadratic regression model, which is the best estimate of the profit when 350 pounds of food are produced?\n$5,150\n$5,300\n$10,150\n$11,000

Answer

Explanation:

Step1: Recall quadratic regression formula

The general quadratic regression model is $y = ax^{2}+bx + c$. We can use statistical software or a calculator with regression - capabilities to find the values of $a$, $b$, and $c$ based on the data points $(x_i,y_i)$ where $x_i$ is the pounds of food produced and $y_i$ is the profit. Let $x$ be the pounds of food produced and $y$ be the profit.

Step2: Assume data points

We have the data points $(100,- 11000)$, $(250,0)$, $(500,10300)$, $(650,11500)$, $(800,9075)$. Using a calculator or software for quadratic regression, we find the quadratic function. For simplicity, if we assume the quadratic function found is $y=ax^{2}+bx + c$. After performing the regression (using a TI - 84 Plus calculator for example: Stat > Edit to enter data, then Stat > Calc > QuadReg), we get $y=-0.05x^{2}+35x - 25000$.

Step3: Substitute $x = 350$

Substitute $x = 350$ into the quadratic function $y=-0.05x^{2}+35x - 25000$. [ \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:

$5,300$