EX 3.36 (EX 3.35 continued)

(EX 3.35 animal population continued)

• In the animal-tagging example:
• Parameters are defined as:
  • $n=10$: number of animals tagged.
  • $M=5$: number of tagged animals.
  • $N=25$: total number of animals.
• Calculate probability:
  • $p=\frac{M}{N}=\frac{5}{25}=0.2$.
• Expected value of $X$:
  • $E(X)=n\cdot p=10\cdot 0.2=2$.
• Variance of $X$ (without replacement):
  • $V(X)=\frac{M}{N-1}\cdot n\cdot p\cdot (1-p)$.
• Substituting values:
  • $V(X)=\frac{15}{24}\cdot 10\cdot 0.2\cdot 0.8=(0.625)(1.6)=1$.
• If the sampling had been carried out with replacement:
  • Variance becomes:
    • $V(X)=1.6$
• Given:
  • Population size $N$ is unknown
  • Observed value $x$ is provided
  • Objective:
    • Estimate the population size $N$
  • Reasoning:
    • Equate observed sample proportion $\frac{x}{n}$ with population proportion $\frac{M}{N}$
  • Conclusion:
    • Provides an estimate for $N$

$\hat{N}=\frac{M\cdot n}{x}$

If $M=100$ , $n=40$, and $x=16$, then $\hat{N}=250$.