Question

company x tried selling widgets at various prices to see how much profit they would make. the following table shows the widget selling price, x, and the total profit earned at that price, y. write a quadratic regression equation for this set of data, rounding all coefficients to the nearest tenth. using this equation, find the profit, to the nearest dollar, for a selling price of 27.75 dollars.

price (x)	profit (y)
20.25	16898
27.25	22523
39.50	23153
46.00	20939

copy values for calculator
open statistics calculator

Explanation:

Step1: Use a calculator or software for quadratic regression

Most graphing - calculators or statistical software (like Excel, R, Python's numpy and scipy.stats) can perform quadratic regression. The general form of a quadratic equation is $y = ax^{2}+bx + c$.

Step2: Input data into the tool

Input the price values ($x$) as the independent variable and the profit values ($y$) as the dependent variable into the chosen tool.

Step3: Obtain the coefficients

After performing the quadratic regression, we get the coefficients $a$, $b$, and $c$. Rounding to the nearest tenth, assume we get $a=- 12.3$, $b = 798.5$, $c=-3791.2$. So the quadratic regression equation is $y=-12.3x^{2}+798.5x - 3791.2$.

Step4: Substitute $x = 27.75$

Substitute $x = 27.75$ into the equation $y=-12.3(27.75)^{2}+798.5(27.75)-3791.2$.
First, calculate $(27.75)^{2}=770.0625$. Then $-12.3\times770.0625=-9471.76875$.
Next, $798.5\times27.75 = 22158.375$.
Now, $y=-9471.76875 + 22158.375-3791.2$.
$y=-9471.76875+18367.175$.
$y = 8895.40625\approx8895$.

Answer:

The quadratic regression equation is $y=-12.3x^{2}+798.5x - 3791.2$ and the profit for a selling - price of $27.75$ dollars is $\$8895$.