Indiv | height | weight |
---|---|---|
1 | 32 | 25 |
2 | 24 | 20 |
3 | 28 | 26 |
4 | 20 | 13 |
5 | 36 | 33 |
6 | 25 | 26 |
Relevant Wildlife/Fish/Habitat studies???
We take morphometric data on all Lowland Tree Kangaroos in a forest of Papua New Guinea (6 individuals).
However, what if you end up releasing 2 individuals before getting their weight, but all individuals height was measured.
What was the mean weight of all individuals in the forest?
Indiv | height | weight |
---|---|---|
1 | 32 | 25 |
2 | 24 | 20 |
3 | 28 | 26 |
4 | 20 | 13 |
5 | 36 | 33 |
6 | 25 | 26 |
\(\mu_{weight} =\) 23.8333333
\(\mu_{height} =\) 27.5
Weight
Sample.Number | Indiv.1 | Indiv.2 | Indiv.3 | Indiv.4 | means |
---|---|---|---|---|---|
1 | 25 | 25 | 25 | 25 | 25.00 |
2 | 25 | 25 | 25 | 25 | 25.00 |
3 | 25 | 25 | 20 | 20 | 22.50 |
4 | 20 | 20 | 26 | 20 | 21.50 |
5 | 20 | 20 | 20 | 20 | 20.00 |
6 | 20 | 26 | 26 | 26 | 24.50 |
7 | 13 | 26 | 26 | 26 | 22.75 |
8 | 13 | 13 | 26 | 26 | 19.50 |
9 | 26 | 13 | 13 | 33 | 21.25 |
10 | 13 | 13 | 33 | 33 | 23.00 |
11 | 13 | 13 | 33 | 33 | 23.00 |
12 | 33 | 13 | 33 | 26 | 26.25 |
13 | 33 | 26 | 26 | 33 | 29.50 |
14 | 26 | 26 | 26 | 33 | 27.75 |
15 | 26 | 26 | 26 | 26 | 26.00 |
\(E[\hat{\mu}_{weight}] =\) 23.8333333 \(=\mu_{weight}\)
Height
Sample.Number | Indiv.1 | Indiv.2 | Indiv.3 | Indiv.4 | means |
---|---|---|---|---|---|
1 | 32 | 32 | 32 | 32 | 32.00 |
2 | 32 | 32 | 32 | 32 | 32.00 |
3 | 32 | 32 | 24 | 24 | 28.00 |
4 | 24 | 24 | 28 | 24 | 25.00 |
5 | 24 | 24 | 24 | 24 | 24.00 |
6 | 24 | 28 | 28 | 28 | 27.00 |
7 | 20 | 28 | 28 | 28 | 26.00 |
8 | 20 | 20 | 28 | 28 | 24.00 |
9 | 28 | 20 | 20 | 36 | 26.00 |
10 | 20 | 20 | 36 | 36 | 28.00 |
11 | 20 | 20 | 36 | 36 | 28.00 |
12 | 36 | 20 | 36 | 25 | 29.25 |
13 | 36 | 25 | 25 | 36 | 30.50 |
14 | 25 | 25 | 25 | 36 | 27.75 |
15 | 25 | 25 | 25 | 25 | 25.00 |
\(E[\hat{\mu}_{height}] =\) 27.5 \(=\mu_{height}\)
\(\hat{\mu}_{r}\) = sample ratio \(\times\) population mean of aux
\(\hat{\mu}_{r}\) = sample primary mean / sample aux. mean \(\times\) population mean of aux . . .
\(\hat{\mu}_{r}\) = \(r \times \mu_{x}\)
\(\mu_{x} = \frac{\sum_{i=1}^N x_i}{N}\)
\(\hat{r} = \frac{\sum_{i=1}^n y_i}{\sum_{i=1}^n x_i} = \frac{\hat{\mu}_{primary}}{\hat{\mu}_{secondary}} =\frac{\hat{\mu}_{weight}}{\hat{\mu}_{height}}= \frac{\bar{y}}{\bar{x}}\)
\[ \hat{\sigma}^2_{\hat{\mu_r}} = \left(\frac{N-n}{N}\right)\frac{\hat{\sigma}^2_r}{n} \]
\[ \hat{\sigma}^2_r = \frac{1}{n-1}\sum_{i=1}^n\left(y_i-rx_i\right)^2 \]
Sample.Number | Primary.Weight | Aux.Height | Ratio |
---|---|---|---|
1 | 32.00 | 25.00 | 21.48438 |
2 | 32.00 | 25.00 | 21.48438 |
3 | 28.00 | 22.50 | 22.09821 |
4 | 25.00 | 21.50 | 23.65000 |
5 | 24.00 | 20.00 | 22.91667 |
6 | 27.00 | 24.50 | 24.95370 |
7 | 26.00 | 22.75 | 24.06250 |
8 | 24.00 | 19.50 | 22.34375 |
9 | 26.00 | 21.25 | 22.47596 |
10 | 28.00 | 23.00 | 22.58929 |
11 | 28.00 | 23.00 | 22.58929 |
12 | 29.25 | 26.25 | 24.67949 |
13 | 30.50 | 29.50 | 26.59836 |
14 | 27.75 | 27.75 | 27.50000 |
15 | 25.00 | 26.00 | 28.60000 |
\(E[\hat{\mu}_{r}] =\) 23.8683977 \(\neq\) 23.83333
Ratio Estimator
Estimating total population
\[ \frac{m_2}{n_2} = \frac{n_1}{N} \] . . .
Lincoln-Peterson Abundance Estimator
\[ \hat{N} = \frac{n_1n_2}{m_2} \] . . .
\[ \hat{N} = \frac{n_1}{\hat{p}} \] \[ \hat{p} = \frac{m_2}{n_2} \]