To estimate the value of [tex]\( y \)[/tex] for [tex]\( x = 4 \)[/tex] based on the given data, we can use polynomial regression. The data points provided are [tex]\( (x, y) \)[/tex]:
[tex]\[
(2.5, 6.25), (9.4, 88.36), (15.6, 243.63), (19.5, 380.25), (25.8, 665.64)
\][/tex]
Here are the steps to determine the approximate value of [tex]\( y \)[/tex] at [tex]\( x = 4 \)[/tex]:
1. Fit a Quadratic Polynomial:
To fit a quadratic polynomial to the given data points, we determine the coefficients [tex]\((a, b, c)\)[/tex] of the quadratic equation [tex]\(y = ax^2 + bx + c\)[/tex].
2. Find the Polynomial Coefficients:
The coefficients obtained for the best-fit quadratic polynomial to these data points are:
[tex]\[
a = 0.99900978, \quad b = 0.02863292, \quad c = -0.08883057
\][/tex]
3. Construct the Polynomial Equation:
Using these coefficients, the quadratic polynomial equation becomes:
[tex]\[
y = 0.99900978x^2 + 0.02863292x - 0.08883057
\][/tex]
4. Calculate [tex]\( y \)[/tex] for [tex]\( x = 4 \)[/tex]:
Substitute [tex]\( x = 4 \)[/tex] into the polynomial equation:
[tex]\[
y = 0.99900978(4)^2 + 0.02863292(4) - 0.08883057
\][/tex]
5. Perform the Calculation:
[tex]\[
y = 0.99900978 \times 16 + 0.02863292 \times 4 - 0.08883057
\][/tex]
[tex]\[
y = 15.98415648 + 0.11453168 - 0.08883057
\][/tex]
[tex]\[
y \approx 16.0098576
\][/tex]
We find that the approximate value of [tex]\( y \)[/tex] for [tex]\( x = 4 \)[/tex] is:
[tex]\[
\boxed{16}
\][/tex]
