STATISTICS for bio

STATISTICS 1403

PROBABILITY AND STATISTICS FOR THE BIOSCIENCES

PROBLEM SET: QUADRAT SAMPLING

Abstract. How many trees on this ranch?

1. Introduction

A Central Texas rancher has recently acquired a 100-hectare (247 acre) tract of grassland which is lightly wooded. He consults a state forester about determining the number of trees on the property. The forester suggests estimating the tree count by random sampling of one-hectare quadrats. If he had a recent aerial photograph (which he doesn’t) the forester would see the trees distributed like this:

1

2 QUADRAT SAMPLING

2. Sampling Plans

The forester considers three sampling plans:

Simple Random Sampling (SRS): He numbers the quadrats from 1 to N = 100. He then draws some smaller number n random numbers in the range 1 . . . 100 without replacement, and counts the trees (yi) in each randomly selected quadrat. From the sample data, he makes a point and interval estimate for the total number of trees as

ȳ = 1

n

n∑ i=1

yi T̂ = Nȳ

s2y =

∑n i=1 y

2 i − nȳ

2

n − 1 T = T̂ ± 1.96N

√ s2y n

( N − n N

) Systematic Random Sampling (SYS): The forester doesn’t want to

generate a lot of random numbers. He numbers the quadrats as before,

and then picks a sample size n and a sampling interval k, where k = N

n .

He then picks one random number r, with 1 ≤ r ≤ k, and samples every kth quadrat, r, r + k, r + 2k, . . .. He estimates the total number of trees using the same formulas as he would with the simple random sample.

Stratified Random Sampling (ST): Based on an initial walk-though, the forester notices that trees seem denser in the northern (green) part of the tract, and sparser in the southern (blue) part. He divides the tract into h = 2 parts, with N1 = 60 and N2 = 40 quadrats in each. He performs a simple random sample for each part, with sample sizes n1 and n2, and estimates the total as

T̂ =

h∑ j=1

T̂j =

h∑ j=1

Njȳj T = T̂ ± 1.96

√√√√ h∑ j=1

N2j

( s2j nj

)( Nj − nj

Nj

) 3. The Samples

Estimate the number of trees using each of the sampling plans above. For the SRS and SYS plans, use a sample size of n = 10 quadrats. For the ST sample, use n1 = n2 = 5. For each sample, provide a table showing the quadrats sampled and the number of trees in each quadrat, and, of course, the point and interval estimates of the total. For example:

region quadrat trees north 2 9 · · · · · · · · · south 68 6

4. Your Analysis

Your analysis should be handwritten. Include a brief problem statement (in your own words, do NOT copy these instructions), and a brief narrative explaining your calculations and findings. Submit your write-up via Blackboard as a scanned or saved .PDF or Word .DOC file.