FR 3218/5218 Spring 2007

Due February 7, 2007

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 Grader 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). You may also use a statistical package if you wish. If you do, turn in printouts that are annotated so that the Grader can follow the calculations. All data for this assignment are in the Excel workbook hw3Data_07.xls.

1. A soil scientist wants to estimate the number of windthrow pits (cavity at base of fallen trees) per acre for a forest regeneration study in a 300-acre forest tract that experienced storm damage. There are three forest types within the 300-acre tract. The northern hardwood type covers 150 acres, the spruce-fir type covers 90 acres, and the pine type covers 60 acres. The scientist suspects that the number of windthrow pits varies among forest types and has decided to use a stratified random sample. The scientist randomly identifies and samples 40 (1/10-acre) plots in the tract (20 in the northern hardwood type, 12 in the spruce-fir type, and 8 in the pine type). Sample data are in the "Windthrow Pits" worksheet. Note that the observations are pits per acre.

1. Estimate the mean number pits per acre for each stratum and obtain associated standard errors.
   
2. Estimate the mean number of pits per acre for the entire 300 acres and provide a 90% confidence interval for it.
   
3. Was stratified sampling sensible in the example? Why or why not? (consider our discussion of under what conditions stratified sampling excels)
   

2. Referring back to the previous question (using those data to estimate variances):

1. Compute the total number of plots that would need to be observed, as well as the number of plots by stratum, under proportional allocation to achieve a desired standard error of 2 pits per acre.
   
2. Compute the total number of plots that would need to be observed, as well as the number of plots by stratum, under optimal allocation (assuming equal costs) to achieve a desired standard error of 2 pits per acre.
   
3. Briefly discuss why the two allocations lead to sampling different total number of plots and plots by stratum allocations.
   

3. Interest lies in the proportion of forest in a particular condition. A recent digital map of the entire forest is available where every satellite pixel has been assigned a proportion value based on a computer algorithm. The digital map suggests 35 percent of the forest is in the particular condition. You decide to take 50 plots in the forest to observe condition for yourself on the ground. You randomly locate your plots on pixel centers. So for those 50 plots you have a ground-based observation and a satellite-based observation (from the digital map). Assume N is VERY LARGE (no FPC). Those data are in the "Proportions" worksheet.

1. Compute the ratio-of-means estimate of the proportion of forest in the condition of interest and provide the standard error of the estimate.
   
2. Compute the regression estimate of the proportion of forest in the condition of interest and provide the standard error of the estimate.
   
3. Which estimator seems most appropriate (if either)? Explain what you base your answer on?