The volume (i.e., the effective wood production in cubic meters), height (in meters), and diameter (in meters) (measured at 1.37 meter above the ground) are recorded for 31 black cherry trees in the...

The volume (i.e., the effective wood production in cubic meters), height (in meters), and diameter (in meters) (measured at 1.37 meter above the ground) are recorded for 31 black cherry trees in the Allegheny National Forest in Pennsylvania. The data are listed in Table 17.3. They were collected to find an estimate for the volume of a tree (and therefore for the timber yield), given its height and diameter. For each tree the volume y and the value of x = d2h are recorded, where d and h are the diameter and height of the tree. The resulting points (x₁, y₁),...,(x₃₁, y₃₁) are displayed in the scatterplot in Figure 17.13. We model the data by the following linear regression model (without intercept)

a. What physical reasons justify the linear relationship between y and d2h? Hint: how does the volume of a cylinder relate to its diameter and height?

 b. We want to find an estimate for the slope β of the line y = βx. Two natural candidates are the average slope ¯zn, where zi = yi/xi, and the