MATH 313: Survey Design and Sampling
\[ \hat{\mu} = \left(\frac{M}{\bar{N}}\right) \frac{\sum_{i=1}^{m} n_i \bar{y}_i}{m} \]
Understanding population size is crucial in fields like ecology, event management, and quality control. This chapter explores various methods for estimating population sizes across different scenarios.
Direct sampling involves tagging and recapturing individuals from a wildlife population to estimate the total population size using the formula: \[ \hat{N} = \frac{t}{\hat{p}} \] where \(t\) is the number of tagged individuals and \(\hat{p}\) is the proportion of tagged individuals observed in the second sample. This method is foundational in ecological studies where precise population counts are needed.
Inverse sampling is similar to direct sampling but does not predetermine the size of the second sample. Instead, sampling continues until a preset number of tagged individuals is recaptured. This method provides an alternative and often more flexible approach to estimating population size: \[ \hat{N} = \frac{t}{\hat{p}} \]
When individuals are not mobile or are evenly distributed, density estimation via quadrat samples becomes effective. By randomly selecting plots and counting individuals within them, density per unit area is estimated and scaled up to the entire area: \[ \hat{\lambda} = \frac{\sum_{i=1}^{m} n_i}{m \cdot a}, \quad \hat{N} = \hat{\lambda} \times A \] where \(\hat{\lambda}\) is the estimated density, \(m\) is the number of sampled plots, \(n_i\) is the count in plot \(i\), and \(A\) is the total area.
This method adapts the traditional density estimation by focusing on the presence or absence of the species rather than exact counts. Useful in scenarios where counting exact numbers is impractical: \[ \hat{\lambda} = -\frac{1}{a} \ln \left(\frac{y}{m}\right), \quad \hat{N} = \hat{\lambda} \times A \] where \(y\) is the number of unstocked quadrats, and \(a\) is the area of each quadrat.
Adaptive sampling involves adjusting the sampling strategy based on initial findings. This technique is especially beneficial in heterogeneous environments where certain areas may exhibit significantly higher density than others: \[ \hat{N} = \hat{\mu} \times M, \quad \hat{\mu} = \frac{1}{m} \sum_{i=1}^{m} \frac{y_i}{m_i} \] where \(m_i\) is the number of additional cells sampled around the initial cell if a species is found, enhancing the estimate’s accuracy.
To estimate the population of a bird species, a conservation team uses direct sampling by tagging and recapturing birds. The initial sample sizes and recapture rates provide the data needed to estimate the total population.
Wildlife Conservation: Using direct sampling, the team estimates a bird species’ population at 1200 based on recapture rates of tagged birds.
An event organizer estimates attendee numbers using adaptive sampling techniques by initially counting in a small area and adjusting counts based on crowd density observed in various venue areas.
Event Attendance: By employing adaptive sampling at a music festival, organizers estimate 25,000 attendees, adjusting their counts based on varying crowd densities across different sections of the venue.