Day 35

MATH 313: Survey Design and Sampling

Introduction to Inverse Sampling

Inverse sampling is a statistical method used to estimate the total population size, \(N\), when the number of tagged and recaptured individuals is a key variable.

Inverse Sampling Method

  • Procedure: Start with an initial tagged sample of size \(t\). Continue sampling until \(s\) tagged individuals are recaptured.
  • Estimation Formula: \[ \hat{N}_I = \frac{n t}{s} \] Here, \(n\) is the number of individuals in the second sample, and \(s\) is fixed while \(n\) is random.

Estimator and Variance

  • Population Size Estimator: \[ \hat{N}_I = \frac{n t}{s} \]
  • Variance of Estimator: \[ \hat{V}(\hat{N}_I) = \frac{t^2 n (n-s)}{s^2 (s+1)} \] This formula assumes \(s > 0\) and provides an unbiased estimation of \(N\).

Confidence Interval Calculation

  • Critical Value and Error Bound: \[ \lambda = \sqrt{\frac{4}{9 \alpha}} \] \[ B_I = \lambda \cdot \sqrt{\hat{V}(\hat{N})} \]
  • Confidence Interval: \[ \hat{N}_I \pm B_I \] Provides a conservative estimate of the population size within a specific confidence level.

Example 1: Authorities of a large wildlife preserve are interested in the total number of birds of a particular species that inhabit the preserve. A random sample of \(t=150\) birds is trapped, tagged, and then released. In the same month, a second sample is drawn until 35 tagged birds are recaptured ( \(s=35\) ). In total, 100 birds are recaptured in order to find 35 tagged ones. Estimate \(N\) and place a \(95 \%\) bound on the error of estimation.