FR 3218/5218 Spring 2008

Due March 10, 2008

This assignment covers a) two-stage sampling, b) developing a local volume equation and c) summarizing data from fixed-radius plot sampling.

It is important that you show your work. Credit cannot be given if answers are not accompanied by intermediate calculations that demonstrate how you arrived at the final solution. The TA will look to the intermediate calculations to determine the level of partial credit to be granted for wrong answers. If you turn in a printout of a spreadsheet be sure to include the formulas you used to calculate all intermediate and final estimates (see Excel insert in the solution set for Assignment 1); better yet, e-mail the spreadsheet file to the TA. You may also use a statistical package if you wish. If you do use a statistical package, turn in printouts that are annotated so that the TA can follow the calculations. All data for this assignment are in the Excel workbook hw4DAT_08.xls.

As a reminder, the last sections in Chapter 2 of your text review regression and you have the Regression Refresher and Excel regression example documents to refer to as well.

1. Interest lies in the number of individuals of a "rare furbearer" in northern Alaska, above the Arctic Circle. The only way to estimate numbers is via extensive ground surveys in the winter, following tracks in the snow. It has been decided that a random sample of 25 townships (Public Land Survey townships) will be taken and then a random sample of three sections (Public Land Survey sections) within selected townships surveyed on the ground. There are a total of 5000 townships in the area of interest. Data are in the worksheet Animals.

1. Estimate the total number of individuals in the area of interest. Provide a standard error for the estimate of total number of individuals.
   
2. Provide an 80% confidence interval for the total number of individuals in the area of interest.
   
3. What characteristics of this problem, do you suppose, led to the selection of a two-stage design over simple random sampling of sections directly. What needs to be the case for the two-stage sample to approach the statistical efficiency of simple random sampling of 75 sections?
   

2. As part of a stand assessment, tree height and DBH data were collected on 38 red pine trees (worksheet HtDBH). You need to develop a local volume equation for the stand using these data and the volume ratio system on pages 177-178 of your text (just assume this volume ratio system will work for your red pine). The volume of interest is to a 5-inch o.b. top diameter. For each DBH (inches), total height (feet) pair in the worksheet, compute a tree volume. Then fit the following two simple linear regression equations to the resulting data.

V = b₀ + b₁ D + error
V = c₀ + c₁ D² + error

where V is volume (what you computed using the volume ratio system), D is DBH, and {b₀, b₁, c₀, c₁} are regression coefficients to be estimated. Write down each fitted equation (identifying each variable) and identify the coefficient of determination and standard error of estimate fit statistics for each equation. Plot standardized residuals versus predicted values (you will need to request the standardized residuals when you specify the regression in Excel and then plot them against predicted values using Excel's Chart Wizard) for each equation separately. Do either or both of the residual plots suggest a problem? Explain. Which equation would you use as a local volume equation for the stand, if either?

3. Eleven (11) trees were "in" on a 1/20-acre fixed-radius plot. The tally results are in worksheet FixedPlot. Compute plot observations of trees/acre, basal area/acre, and volume per acre. The volume of interest is board feet based on Table 2 of Bulletin 1104 (handed out in lecture).

4. Twelve (12), 1/15-acre plots were tallied in a mixed-species stand. The cumulative tally across all 12 plots is given in worksheet CumuTally. Estimate stand (trees per acre by DBH class and species) and stock (volume per acre by DBH class and species) tables from the cumulative tally. The volume of interest is cords based on Table 6 of Bulletin 1104 (handed out in lecture). Be sure to consult the examples of stand and stock tables handed out in class to format and present the tables properly.