Question

the table shows the number of hats a company can sell at different prices. use quadratic regression to determine how many hats the company will sell if it charges $9 for each hat.
price (dollars), x: 3 4 5 6 7
hats sold (thousands), y: 126 144 150 144 126

Explanation:

Step1: Set up quadratic regression model

The general quadratic - regression equation is $y = ax^{2}+bx + c$. We have a system of equations based on the data points. For $(x_1,y_1)=(3,126)$, $(x_2,y_2)=(4,144)$, $(x_3,y_3)=(5,150)$, $(x_4,y_4)=(6,144)$, $(x_5,y_5)=(7,126)$. Substituting into $y = ax^{2}+bx + c$ gives:
$126 = 9a+3b + c$
$144 = 16a+4b + c$
$150 = 25a+5b + c$
$144 = 36a+6b + c$
$126 = 49a+7b + c$
We can also use a statistical software or a calculator with regression capabilities. Using a calculator (TI - 84 Plus for example):
Enter the $x$ - values in list $L_1$ and the $y$ - values in list $L_2$. Then perform quadratic regression. The quadratic regression formula is $y=-6x^{2}+60x + 36$.

Step2: Substitute $x = 9$ into the regression equation

Substitute $x = 9$ into $y=-6x^{2}+60x + 36$.
$y=-6\times9^{2}+60\times9 + 36$
$y=-6\times81+540 + 36$
$y=-486+540 + 36$
$y = 90$

Answer:

90000 (since $y$ is in thousands, the number of hats is $90\times1000 = 90000$)