Resource assessment exercises: estimating a proportion
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## [1] 0.68 | ## [1] 0.68 | ||
</pre> | </pre> | ||
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| + | Indexing using the double equal sign \texttt{==}: the double equal sign is used to access | ||
| + | entries that exactly match a condition. Above we select all entries \texttt{s.species} for | ||
| + | which \texttt{s.species == 2} is true. Other operators include \verb|!=| (not equal to), | ||
| + | \verb|<| (smaller than), \verb|>| (larger than), \verb|<=| (smaller or equal to), \verb|>=| | ||
| + | (larger or equal to). | ||
We estimated that 68% of the trees in the population are beech trees. The estimated number of oak trees in the population is <math>\hat{q}=1-\hat{p}</math>. The standard error of the estimated proportion <math>\hat{p}</math> is given by, | We estimated that 68% of the trees in the population are beech trees. The estimated number of oak trees in the population is <math>\hat{q}=1-\hat{p}</math>. The standard error of the estimated proportion <math>\hat{p}</math> is given by, | ||
Revision as of 13:02, 23 June 2014
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This section is still under construction! This article was last modified on 06/23/2014. If you have comments please use the Discussion page or contribute to the article! |
As mentioned in the introduction there are two tree species in the example population: beech trees and oak trees. Suppose we would like to estimate the proportion of beech trees by looking at a sample of \(n=50\) again.
s.species <- sample(trees$species, size = 50) s.species ## [1] 2 1 2 1 2 2 2 2 2 2 2 1 2 2 2 1 2 2 2 2 2 2 2 1 1 2 1 2 2 1 2 1 1 1 1 1 1 2 ## [39] 2 2 2 2 2 2 2 2 2 1 1 2
The 1s are oak trees and the 2s are beech trees. The proportion of beech trees is estimated by
| $\hat{p}=\frac{y_{i\in s} }{n}\quad\text{where}\quad y_i = \left\{ \begin{array}{l l} 1 & \quad \text{if} \,i\, \text{is a beech tree}\\ 0 & \quad \text{otherwise} \end{array} \right.$ | 1 |
In R:
n.beech <- length(s.species[s.species == 2]) n.beech ## [1] 34 p <- n.beech/n p ## [1] 0.68
Indexing using the double equal sign \texttt{==}: the double equal sign is used to access entries that exactly match a condition. Above we select all entries \texttt{s.species} for which \texttt{s.species == 2} is true. Other operators include \verb|!=| (not equal to), \verb|<| (smaller than), \verb|>| (larger than), \verb|<=| (smaller or equal to), \verb|>=| (larger or equal to).
We estimated that 68% of the trees in the population are beech trees. The estimated number of oak trees in the population is \(\hat{q}=1-\hat{p}\). The standard error of the estimated proportion \(\hat{p}\) is given by,
| $s_{\hat{p} }=\sqrt{\frac{\hat{p}\times\hat{q} }{n-1} }$ | 2 |
In R:
q <- 1 - p sqrt((p * q)/(n - 1)) ## [1] 0.06664
Confidence intervals can be constructed in the same way as for the mean estimator above.
Since we know the species of each tree in the population, we can calculate the true proportion of beech trees,
nrow(trees[trees$species == 2, ])/nrow(trees) # true proportion of beech trees ## [1] 0.6702
