lecture 3 forestry 3218 avery and burkhart, chapter 3 shiver and borders, chapter 2 forest...
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Lecture 3Forestry 3218
Avery and Burkhart, Chapter 3Shiver and Borders, Chapter 2
Forest Mensuration II
Lecture 3Elementary Sampling Methods: Selective, Simple Random, and Systematic
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Lecture 3Forestry 3218
Sampling vs. Complete Enumeration
Why sampling?• Measuring all units
(trees, birds, etc.) is sometimes impractical, if not impossible– Some measurements are
destructive
• Sampling saves money and time
Complete Enumeration• Measure every feature
of interest; a highly accurate description of the population.
• Drawbacks: only viable with small populations; only cost-effective with high-valued features.
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Lecture 3Forestry 3218
Sampling Design
• The method of selecting non-overlapping sample units to be included in a sample
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Lecture 3Forestry 3218
Sampling Frame
• The list of all possible sampling units that might be drawn in a sample
• Developing a reliable frame may be difficult – Jack pine trees in Crown forest (infinite population)– In most field situation, differences between the
sampling frame and the population are inconsequential
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Lecture 3Forestry 3218
Elementary Sampling Methods
• Selective• Simple Random
Sampling• Systematic Sampling
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Lecture 3Forestry 3218
Selective Sampling
• The method involved selecting areas that appeared to be reprehensive of the average stand condition to the sampler (cruiser)
• Was widely used in forestry, is still…• Depends on skill of the cruiser, biased• No valid variance, and therefore no confidence
interval, could be calculated• Because sampled areas appeared to be
average, their variability would be smaller than the true variability
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Lecture 3Forestry 3218
Simple Random Sampling (SRS)
• Sampling units are chosen completely at random
• Every possible combination of sampling units has an equal and independent chance of being selected
• SRS is the fundamental method for other sampling procedures
• Other procedures are simply modifications to achieve better precision or greater economy
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Lecture 3Forestry 3218
SRS Procedure
• Requires the development of a frame, implying the need of aerial photographs, or maps
• Select random numbers between one and the total number of sampling units in the population
• Samples are either chosen with replacement or without replacement, the latter means that once a sampling unit is chosen it may not been chosen again
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Lecture 3Forestry 3218
SRS Estimators
xsCV
Mean
Variance
Coefficient of variation
1
2)(2
nxxs
nxx
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Lecture 3Forestry 3218
SRS Estimators
• Standard error of the mean– With replacement or
infinite population– without replacement
from a finite population
• Confidence limit
)(2
N
nNs
n
sx
n
sxs
2
xstx
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Lecture 3Forestry 3218
Sampling Intensity
• How many samples to take? Depends on: – The variability of the population– Desired confidence interval– Acceptable level of error
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Lecture 3Forestry 3218
Sampling Intensity
• With replacement or infinite population
• Without replacement from a finite population
2
E
stn
NstE
n1
21
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Lecture 3Forestry 3218
Calculating sample size
2
E
stn
Standard deviation (120 m3/ha)
95% confidence (t=2)
Acceptable level of error
±40 m3/ha
36
2
40
1202
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Lecture 3Forestry 3218
Calculating sample size from CV and A
2
E
stn
22
ACVt
AxxCVt
xCVsxsCV
AxE
1445302
2
)(nExample:
Allowable percent error of mean
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Lecture 3Forestry 3218
Relationship between sample size and allowable error for different CVs
n
5 205 405 605 805
Allo
wab
le e
rror
(%
)
CV=100
CV=20
0
20
10
30
40
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Lecture 3Forestry 3218
Can we use SRS all the time? - problems
• Locating some sample units on the ground may be very time-consuming– Reference point to sample units– Access
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Lecture 3Forestry 3218
Systematic Sampling
The initial sampling unit is randomly selected. All other sample units are spaced at uniform intervals throughout the area sampled
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Lecture 3Forestry 3218
Systematic Sampling
Pros:• Sampling units are easy
to locate• Sampling units appear
to be “representative”• Generally acceptable
estimates for the population mean
Cons:• Impossible to estimate the
variance of one sample• Accuracy can be poor
(i.e., bias) if a periodic or cyclic variation inherent in the population
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Lecture 3Forestry 3218
Arguments of systematic sampling
Against– SRS statistical techniques can’t logically be
applied to a systematic design unless populations are assumed to be randomly distributed
For– There is no practical alternative to assuming that
populations are distributed in a random order
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Lecture 3Forestry 3218
Summary for Systematic Sampling
• Use systematic sampling to obtain estimates about the mean of populations
• Numerical statement of precision should be viewed as an approximation
• Use SRS formulas
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Lecture 3Forestry 3218
Summary
• Selective sampling• SRS• Systematic sampling