FR 3218/5218 Spring 2007

Due March 5, 2007

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 use 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). You may also use a statistical package if you wish. If you do, turn in printouts that are annotated so that the TA can follow the calculations. All data for this assignment are in the Excel workbook hw4Data_07.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. The manager of a park is interested in estimating the number of skiers using the park's new 25K trail. The period of interest covers 90 days in the winter when the trail is groomed. The manager decides to use a two-stage sampling design. Fifteen days are sampled at random. For each day, 2 "time blocks" are sampled at random. Each block is a 2.5-hour period of time; there are 5 time blocks total each day (12.5 hours of skiing a day). The numbers of skiers are observed for each time block sampled. Data are in the worksheet Skiers.

1. Estimate the total number of skiers on the trail for the 90 day period of interest. Provide the standard error of the estimate.
   
2. Provide a 95% confidence interval for the total number of skiers during the period of interest.
   
3. Briefly describe the advantage of this two-stage design over simple random sampling of blocks-of-time directly. What needs to be the case for the two-stage sample to approach the statistical efficiency of simple random sampling of 30 time blocks?
   

2. As part of a stand assessment, tree height and DBH data were collected on 35 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 2-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. Sixteen (16) trees were "in" on a 1/15-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. Fifteen (15), 1/20-acre plots were tallied in a mixed-species stand. The cumulative tally across all 15 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.