12.1 Capture - Recapture Sampling
Think About It!
In a free concert given on the Old Main lawn, we want to estimate the number of attendees. How are you going to conduct a sampling for this purpose?
ANSWER: At the beginning of the concert, 500 Penn State t-shirts were randomly given out to attendees. 200 attendees are randomly sampled and we find that 40 have the Penn State t-shirt.
Application Exercise
How many total attendees are at the concert using values given in the answer to the 'Think About It' question above?
ANSWER:
\(\hat{\tau}=\dfrac{y}{x}\cdot X =\dfrac{200}{40}\cdot 500=2500\)
\(\hat{V}ar(\hat{\tau})=\dfrac{500\times 200(500-40)(200-40)}{40^3}=115000\)
\(\hat{S}D(\hat{\tau})=339.16\)
A 95% confidence interval is:
\(2500 \pm1.96 \times 339.16\)
\(2500 \pm 664.67\)
Remark: y can be larger than X.
12.2 Inverse Sampling for Capture-Recapture
Application Exercise
Estimate the total population size of eagles for the above example and find the variance of your estimate.
ANSWER:
X = 200
x = 35, y = 100
\(\hat{\tau}=\dfrac{100}{35}\times 200=571.43\)
\(\hat{V}ar(\hat{\tau})=\dfrac{200^2\times 100(100-35)}{35^2(35+1)}=5895.69\)
\(\hat{S}D(\hat{\tau})=76.78\)
12.3 Random Response Model
Application Exercise
If we survey 400 subjects and the number who answer yes to the composite question is 128, extimate the proportion of people who have falsified their tax return.
ANSWER:
\(n=400\)
\(n_1=128\)
\(\hat{p}=2\left(\dfrac{128}{400}-\dfrac{1}{4}\right)=0.14\)
\(\hat{V}ar(\hat{p})=\dfrac{4}{400}\times \dfrac{128}{400}\times \left(1-\dfrac{128}{400}\right)=0.022\)
\(\hat{s}.d.(\hat{p})=0.047\)