Given the regression equation:

$y_1 = ax_1^2 + bx_1 + c$

with the parameters:

• $a = 2.2381$
• $b = 4.46667$
• $c = 5.82857$

We need to find the value of $y_1$ when $x_1 = 5$.

Substitute $x_1 = 5$ into the equation:

$y_1 = 2.2381(5)^2 + 4.46667(5) + 5.82857$

Let's calculate that.The value of $y_1$ when $x_1 = 5$ is approximately 84.11.

Would you like more details on this calculation or have any other questions?

Here are some related questions you might consider:

1. How would the value of $y_1$ change if $x_1$ was 10?
2. What does the coefficient $R^2 = 0.9957$ indicate about the model?
3. How does the residual $e_1$ relate to the accuracy of the model?
4. What would happen to $y_1$ if the parameter $b$ were decreased?
5. Can you interpret the significance of the coefficients $a$, $b$, and $c$ in this quadratic model?

Tip: The $R^2$ value close to 1 suggests that the model fits the data very well, explaining most of the variance in the dependent variable.