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: Recall quadratic regression formula

The general form of a quadratic function is $y = ax^{2}+bx + c$. We can use a system of equations based on the data points $(x,y)$ to find $a$, $b$, and $c$. For $n$ data - points $(x_i,y_i)$ ($i = 1,\cdots,n$), we can set up the following normal equations:
\(\sum_{i = 1}^{n}y_i=an + b\sum_{i = 1}^{n}x_i+c\sum_{i = 1}^{n}x_i^{2}\)
\(\sum_{i = 1}^{n}x_iy_i=a\sum_{i = 1}^{n}x_i + b\sum_{i = 1}^{n}x_i^{2}+c\sum_{i = 1}^{n}x_i^{3}\)
\(\sum_{i = 1}^{n}x_i^{2}y_i=a\sum_{i = 1}^{n}x_i^{2}+b\sum_{i = 1}^{n}x_i^{3}+c\sum_{i = 1}^{n}x_i^{4}\)
Here $n = 5$, \(\sum_{i = 1}^{5}x_i=3 + 4+5+6+7 = 25\), \(\sum_{i = 1}^{5}y_i=126 + 144+150+144+126 = 690\), \(\sum_{i = 1}^{5}x_i^{2}=3^{2}+4^{2}+5^{2}+6^{2}+7^{2}=9 + 16+25+36+49 = 135\), \(\sum_{i = 1}^{5}x_iy_i=3\times126+4\times144+5\times150+6\times144+7\times126=378+576+750+864+882 = 3450\), \(\sum_{i = 1}^{5}x_i^{3}=3^{3}+4^{3}+5^{3}+6^{3}+7^{3}=27+64+125+216+343 = 775\), \(\sum_{i = 1}^{5}x_i^{4}=3^{4}+4^{4}+5^{4}+6^{4}+7^{4}=81+256+625+1296+2401 = 4659\).
Substituting into the normal equations:
\(690 = 5a+25b + 135c\)
\(3450=25a + 135b+775c\)
\(14820=135a+775b + 4659c\)
We can also use a graph - ing calculator or software. Using a graph - ing calculator (TI - 84 Plus for example):

1. Press STAT, then EDIT. Enter the $x$ - values in L1 and the $y$ - values in L2.
2. Press STAT, then move to CALC, and select QuadReg.
3. The calculator will give the values of $a=-6$, $b = 60$, $c = 30$. So the quadratic regression equation is $y=-6x^{2}+60x + 30$.

Step2: Substitute $x = 9$ into the equation

Substitute $x = 9$ into $y=-6x^{2}+60x + 30$.
\[

$$\begin{align*} y&=-6\times9^{2}+60\times9 + 30\\ &=-6\times81+540+30\\ &=-486+540+30\\ &=84 \end{align*}$$

\]

Answer:

84000 (since $y$ is in thousands, the number of hats sold is 84000)