Thank you for the image. Let's break down and solve the questions from the prompt:

Q1: Building Models for Assumptions (i) and (ii)

The goal is to predict the number of board feet divided by 10 (y) using the diameter of the tree in inches (x).

Assumption (i): Trees are modeled as right-circular cylinders of the same height.

• The volume of a cylinder:
  $V = \pi r^2 h$
  Since $r = \frac{x}{2}$, the equation becomes:
  $V \propto x^2$
  This suggests the relationship between $y$ and $x$ is quadratic:
  $y = a x^2 + b$

We’ll fit a quadratic model for this assumption.

Assumption (ii): Trees are right-circular and their height is proportional to the diameter.

• In this case, height $h \propto x$, so the volume becomes:
  $V = \pi \left(\frac{x}{2}\right)^2 \cdot (kx) \propto x^3$
  Thus, the relationship is cubic:
  $y = a x^3 + b$

We’ll fit a cubic model for this assumption.

Q2: Fitting the Cubic Model using Least Squares and Finding Chebyshev Bounds

• Since assumption (ii) is deemed more realistic, we'll focus on fitting the cubic model:
  $y = a x^3 + b$
  Using the Least Squares Method, we need to determine the best-fitting coefficients $a$ and $b$ that minimize the squared residuals between the actual and predicted $y$-values.

• Chebyshev criterion: The Chebyshev bounds provide a maximum deviation between the observed values and the fitted model. We’ll compute the bounds as part of evaluating the fit.

Next Steps

1. Fit the quadratic and cubic models using the data provided.
2. Evaluate the cubic model using Least Squares and find Chebyshev bounds.

Do you want me to proceed with the calculations and generate the fitted models for you? I can walk through the entire fitting process using the provided data.