Question

In Exercises 14.34-14.43, use the technology of your choice to
a. obtain and interpret the standard error of the estimate.
b. obtain a residual plot and a normal probability plot of the residuals.
c. decide whether you can reasonably consider Assumptions I-3 for regression inferences met by the two variables under consideration.

14.43 Short eaf Pines. The ability to estimate the volume of a tree based on a simple measurement, such as the diameter of the tree, is important to the lumber industry, ecologists, and conservationists. Data on volume, in cubic feet, and diameter at breast height in inches, for 70 shortleaf pines was reported in $\mathrm{C}$. Bruce an F. X. Schumacher's Forest Mensuration (New York: McGraw-Hill 1935) and analyzed by A. C. Akinson in the article "Transforming Both Sides of a Tree" (The American Statistician, Vol. 48, pp. 307-312). The data are provided on the WeissStats site.

Short answer

(a) The standard error of estimate is $9.875$.

(b)The points closely resemble the normal probability plot that obtained from the residual plot and the normal probability plot of the residuals.

(c) For the provided sample, it is not reasonable to consider the assumptions for regression conclusions met for the variables under investigation.

Step-by-step solution

Part (a) Step 1: Given information

To obtain the standard error of the estimate and interpret.

Part (a) Step 2: Explanation

A sample of residences in a certain area's lot sizes and assessed values are provided.
The Residual as follows:
The error is described as $e=\hat{y}-y$, where $\hat{y}$is the projected response variable value and $y$ is the actual response variable value.
The standard error of estimate is calculate as follows:
The standard error of estimation for a collection of $n$observations is given as:

$S_e=\sqrt{\frac{SSE}{n-2}}$
where SSE stands for squared error sum
Then the Residual plot will be computed as follows:

As a result, the standard error of estimate is $9.875$.
The predicted volume differs by $9.875$cubic $ft$on average from the measured volume, which is a point estimate of common standard deviation.

Part (b) Step 1: Given information

To obtain a residual plot and a normal probability plot of the residuals.

Part (b) Step 2: Explanation

Obtain the residual plot of the residuals by using MINITAB as follows:

The output will be:

The residuals, or values of $e$, corresponding to the values of $x$ are plotted on the graph .
The normal probability plot of the residuals by using MINITAB as follows:

The points closely resemble the normal probability plot.

Part (c) Step 1: Given information

To consider Assumptions$1-3$ for regression inferences met by the two variables under consideration.

Part (c) Step 2: Explanation

Let, assumption for regression inferences as follows:

• Population regression line: There are constants ${\beta }_0$and ${\beta }_1$such that the conditional mean of the response variable $y$is ${\beta }_0+{\beta }_{1}x$for each value of the predictor variable $x$.
• Equal standard deviation: The response variable $y$has the same conditional standard deviation for all values of the predictor variable$x$.
• For any value of the predictor variable $x$, the conditional distribution of the response variable $y$is a normal distribution.
• Independent observations: The responses variable's observations are independent to one another.

Let, the assumption for residual analysis for the regression model is considered as follows:

• The residuals should fall roughly in a horizontal band centered and symmetric about the $x$-axis when plotted against the recorded values of the predictor variable.
• The residuals in a normal probability plot should be nearly linear.

The normal probability plot is nearly linear, indicating that the assumption of normality is met.
The residuals plot displays a concave ascending curve with fluctuation in the middle of the $x$ values. So, there is no concept of a shared standard deviation.
As a result, for regression inferences, assumptions $1$and $2$ are violated.
Furthermore, removing an outlier has no effect on the pattern or variance of the data. For the provided sample, it is not feasible to consider the assumptions for regression conclusions met for the variables under investigation.