Question

Foresters are interested in predicting the amount of usable lumber they can harvest from various tree species. They collect data on the diameter at breast height (DBH) in inches and the yield in board feet of a random sample of $20$Ponderosa pine trees that have been harvested. (Note that a board foot is defined as a piece of lumber $12$inches by $12$inches by $1$inch.) A scatterplot of the data is shown below

(a) Some computer output and a residual plot from a least squares regression on these data appear below. Explain why a linear model may not be appropriate in this case.

(B) Use both models to predict the amount of usable lumber from a Ponderosa pine with diameter $30$inches. Show your work.

(c) Which of the predictions in part (b) seems more reliable? Give appropriate evidence to support your choice.

Short answer

a). A linear model is not ideal there are curvature is residual plot.

b). The regression line is option 1. $\hat{y}=117.0899$and option 2. $\hat{y}=102.967$.

c) The predictions in part (b) seems more reliable are option 1: $\hat{y}=117.0899$.

Step-by-step solution

Part (a) Step 1:  Given information

A scatterplot of the data is shown below,

Part (a) Step 2: Explanation

If a linear model is adequate, the histogram should be about normal, and the residuals scatterplot should indicate random scatter. The linear model is not appropriate if the residual plot shows a curved relationship.

Part (b) Step 1: Given Information 

The using all models to estimate the volume of available pine lumber 30 inches in diameter from a ponderosa pine.

Part (b) Step 2: Explanation 

Option 1: Regression line of least square,

$\hat{y}=a+bx$

The coefficients $a$and $b$ are:

$a=2.078$and

$b=0.0042597$

Then the regression line of least square is:

$\hat{y}=2.078+0.0042597x^3$

With $x=DBH$and $y$the yield

Putting the $X$by$30$:

$\hat{y}=2.078+0.0042597(30{)}^3$

$=117.0899$.

Part (b) Step 3: Explanation 

Option 2: Regression line is $\hat{y}=a+bx$

The coefficients $a$and $b$are $a=1.2319$and

$b=0.113417$

Then the regression line of least square is

$\hat{y}=a+bx$

The coefficients $a$and $b$are$a=1.2319$

$b=0.113417$

Then the regression line of least-squares becomes:

With $x=$DBH and $y$the yield.

Putting$x$by $30$: $\hat{y}=102.967$.

Part (c) Step 1: Given Information 

The prediction that is looks more reliable in part (b).

Part (c) Step 2: Explanation 

From the part (b)

Option 1: $\hat{y}=117.0899$

Option 2: $\hat{y}=102.967$

Option 1 provides a more accurate estimate because its residual plot shows no discernible trend, but Option 2's residual plot shows a discernible curve.