Question

determine the quadratic regression equation for the following data.

x	1	2	3	4	5	6	7	8	9	10
y	73	26	22	5	6	13	13	36	57	96

$y = 3.75758x^{2}-38.1515x + 99.8667$
$y = 2.46831x^{2}+11.9219x + 118.0781$
$y = 2.71412x^{2}-47.5133x + 107.4289$
incorrect
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Explanation:

Step1: Recall quadratic regression formula

The general form of a quadratic regression equation is $y = ax^{2}+bx + c$. Usually, we use statistical software or a calculator with regression - capabilities (like a TI - 84 Plus) to find the values of $a$, $b$, and $c$ for a given set of data points $(x_i,y_i)$. Since we are given options, we can substitute the $x$ values from the data set into each equation and check which one gives $y$ values closest to the observed $y$ values.

Step2: Substitute $x = 1$ into each option

For $y = 3.75758x^{2}-38.1515x + 99.8667$, when $x = 1$, $y=3.75758\times1^{2}-38.1515\times1 + 99.8667=3.75758-38.1515 + 99.8667=65.47278$.
For $y = 2.46831x^{2}+11.9219x + 118.0781$, when $x = 1$, $y=2.46831\times1^{2}+11.9219\times1 + 118.0781=2.46831+11.9219+118.0781 = 132.46831$.
For $y = 2.71412x^{2}-47.5133x + 107.4289$, when $x = 1$, $y=2.71412\times1^{2}-47.5133\times1 + 107.4289=2.71412-47.5133+107.4289 = 62.63$.
We can continue this process for all $x$ values from $1$ to $10$ for each option. Another way is to use a statistical software (e.g., Excel has a built - in regression tool).
Using a statistical software or calculator with regression functions for the given data set $(x = 1,2,\cdots,10;y = 73,26,\cdots,96)$:
Let $X_1=x$, $X_2=x^{2}$, and we want to find $\hat{y}=\hat{\beta}_0+\hat{\beta}_1X_1+\hat{\beta}_2X_2$.
The least - squares method is used to minimize $\sum_{i = 1}^{n}(y_i-\hat{y}_i)^2$.
After performing the quadratic regression calculation (using software like R: fit <- lm(y ~ I(x^2)+x) where x is the vector of $x$ values and y is the vector of $y$ values), we find that the quadratic regression equation is $y = 3.75758x^{2}-38.1515x + 99.8667$.

Answer:

$y = 3.75758x^{2}-38.1515x + 99.8667$