Resource assessment exercises: finite population correction
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Finite population correction
If we observe all values \(y_{i\in U}\) we talk about a census. The mean is calculated, not estimated, i.e.,
## [1] 21.05
## [1] 21.05
As noted above, SRSwoR stands for simple random sampling without replacement (woR). However, if we take a sample with replacement (SRS; set replace = TRUE) we get a slightly different value. This would be an estimate.
## [1] 20.96
If we are interested in a population parameter and take an SRSwoR of size \(n=N\), we get the true population value. There is no doubt about its value (assuming that measurement errors are absent). However, if we estimate the standard error for the \(n=N\) sample we get a positive value instead of a zero \(s_{\bar{y}}\).
## [1] 0.07416
This cannot be! The estimator of the standard error holds for sampling with replacement. For sampling without replacement we have to correct for the fact that we took a relatively large sample. The finite population correction (fpc) for a relatively large sample is defined as,
\(\text{fpc}=1-\frac{n}{N}. \tag{1}\)
Obviously, if \(n=N\) the fpc becomes zero. Suppose we take a sample of size \(n=25,000\) from trees, then
## [1] 0.08099
## [1] 0.03307
## [1] 0.03307
For the parametric standard error the fpc becomes,
\(\text{fpc}=\frac{N-n}{N-1}. \tag{2}\)
As a rule of thumb, we apply the fpc when the sampling fraction
\(f=\frac{n}{N} \tag{3},\)
exceeds 0.05, i.e., 5 percent.
Required sample size determination
Above we took a sample S of size \(n=50\).
## [1] 22 8 18 43 21 44 17 25 32 10 11 17 9 10 56 14 14 10 20 8 37 14 55 29 33 ## [26] 17 10 15 29 8 21 9 9 24 21 28 19 58 16 16 15 20 5 9 14 30 11 9 12 27
The width of the confidence interval was,
## [1] 7.364
Suppose a confidence interval of \(A=3\) cm is desired. How large should the sample size, \(n\), be? This can be estimated,
\(A=t_{\alpha,n-1}\frac{s}{\sqrt{n}}\rightarrow n=\frac{t_{\alpha, n-1}^2s^2}{A^2} \tag{4}\)
We will use the sample S (\(n=50\)) to estimate how many observations we need in our sample. In :
## [1] 301.2
We always need to round up!
We estimate the width of the confidence interval using the new sample size of \(n=302\).
## [1] 2.759
