STT 350: Survey sampling Dr. Cuixian Chen Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow STT 350: Survey sampling Dr. Cuixian Chen Chapter 10 Chapter 10: Estimating the Population size

Why to estimate population size N? Previous chapters: estimate means, totals, and proportions, assuming that population size N is either known, or unknown but large. Frequently, N is not known and is important to the goals of the study. In fact, in some studies, estimation of N is the main goal. E.g.: Study of growth, evolution, and maintenance of wildlife populations depends crucially on accurate estimates of N. E.g.: Estimate # of people at a concert or a sporting event, the # of defects in a bolt of material, and many similar quantities.

Population size N estimation methods Five methods: 1) Direct sampling; 2) Inverse sampling; 3) 3rd depends on first estimating density of elements in population and then multiplying by an appropriate measure of area. 4) 4th similar to the third, but depends only on being able to identify presence or absence of animals on the sampled plots. Then under certain conditions the density and the total number of animals can still be estimated. 5) 5th extends density estimation method to make use of adaptive sampling, a technique for adding to sampled units while fieldwork is in process so as to make good use of high-density areas as they are discovered.

Chap 10.2: Ideas of direct sampling Direct sampling: to estimate N of a mobile population. First, a random sample of size t is drawn from population. At a later date a second sample of size n is drawn. E.g., suppose a conservationist is concerned about the apparent decline in # of seals in Alaskan waters. Estimates of population size are available from previous years. For a determination of whether there has been a decline, a random sample of seals is caught, tagged, and then released. A month later a second sample of size is obtained. Using these data (often called recapture data), we can estimate N, population size. This method assumes that tagging does not affect the likelihood of recapture. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 8

Estimate N (pop size) Using Direct Sampling: by Lincoln-Petersen Method Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Eg 10.1, page 315 Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Estimate N (pop size) Using Direct Sampling: by Chapman Method Revisit Example 10.1, by Chapman Method: Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Ex 10.8, page 329 Use both Lincoln-Petersen Method, and Chapman Method. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Chap 10.3: Ideas of Inverse Sampling Inverse sampling is second method for estimating N the total size of a population. We again assume that an initial sample of t individuals is drawn, tagged, and released. Later, random sampling is conducted until exactly s tagged animals are recaptured. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 8

Estimate N (pop size) Using Inverse Sampling Let s be # of tagged individuals observed in second sample. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Estimate N (pop size) Using Inverse Sampling See: Variance for inverse sampling is similar to variance for direct sampling. Estimators appear to be identical. Inverse sampling offers advantages that s can be fixed in advance, is unbiased, and an unbiased estimator of the true variance of is available. Estimators for two procedures are identical; critical difference is that for inverse sampling, s is chosen (i.e., it is not a random value arising from the sampling). Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Eg 10.2, page 317 Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Another Example for inverse sampling A zoologist wishes to estimate the size of the turtle population in a given geographical area. She believes that the turtle population size is between 1000 and 1500; hence, an initial sample of 150 (10%) appears to be sufficient. The t =150 turtles are caught, tagged, and released. A second sampling is begun one month later, and she decides to continue sampling until s=20 tagged turtles are recaptured. She catches 180 turtles before obtaining 20 tagged turtles ( that is, n=180, s=20). Q: Estimate N and place a bound on the error of estimation. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Chap 10.5: Estimating Population Density and Size from Quadrat Samples Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 8

Eg 10.4, page 321 Q: With original data provided, use both EXCEL and R to complete this EX, find the estimate of the density estimation and the bound. x=c(rep(0, 13), rep(1, 8), rep(2, 12), rep(3, 10), rep(4, 5), rep(5, 2)) var(x); #finding the variance; sum(x); # find the sum; mean(x); #finding the mean; sd(x); #finding the SD; length(x); #finding the number of entry in the data; summary(x); #finding the 5-number summary. median(x); #finding the median. hist(x); boxplot(x); Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Ex10.19, page 330 With original data provided, use either EXCEL or R, by both of the following procedures : Q: find the density estimation and bound. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Ex10.19 -- EXTRA Ex, page 330 For the situation and data in Exercise 10.19: Q: Estimate the total number of bacteria colonies in the lab with 2,000 of such fields. Place a bound on the error of estimation. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Ex10.16 page 330 Q: Is the original data provided? Q: Which procedure we should go: Procedure I, or Procedure I, or Both? Q: Perform the data analysis. Q: What is the further assumption under this case? Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Chap 10.6: Estimating Population Density and Size from Stocked Quadrat Samples In quadrat sampling of plants or animals, difficult to count exact # of species under investigation. In contrast, often easier to detect presence or absence of species of interest. Knowing whether a species is present in a sample quadrat can lead to an estimate of density and of population size. Foresters refer to a quadrat that contains the species of interest as being stocked. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 8

Review STT315: Poisson distribution Dr. Cuixian Chen Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Review STT315: Poisson distribution Dr. Cuixian Chen Chapter 10 Chapter 10: Estimating the Population size

3.8 The Poisson Dist, (one of the most important) Idea of Poisson distribution: The Poisson probability distribution often provides a good model for the probability distribution of the number Y, where: Y = rare events that occur in unit space, unit time, unit volume, or any other dimension, where λ is the average value of Y. For large n, small p, and n*p<=7. Examples: (1) automobile accidents in a given unit of time; (2) industrial accidents, or other types of accidents in a given unit of time. (3) number of telephone calls handled by a switchboard in a time interval, (4) the number of radioactive particles that decay in a particular time period, (5) the number of errors a typist makes in typing a page, (6) the number of automobiles using a freeway access ramp in a ten-minute interval.

3.8 The Poisson Dist, (one of the most important) If  is the average number of outcomes per unit time (arrival rate), the Poisson distribution gives the probability that y=outcomes occur in a given time interval(or given area ,etc.). Def: The probability distribution of the Poisson random variable Y, representing the number of rare outcomes occurring in a given time interval or specified region, is where λ is the average number of outcomes in the given, distance, area or volume, and e = 2.71828…… We denote it by Y~Pois (λ). Eg: During a laboratory experiment the average number of radioactive particles passing through a counter in 1 millisecond is 4 1. What is the probability that 6 particles enter the counter in a given millisecond? 2. What is the probability that two or more particles enter the counter in a given millisecond?

3.8 The Poisson dist. Eg3.19: Suppose that a police patrol is visiting a given beat location on average once in a half-hour period. Assume that the number of visits per half-hour follows Poisson distribution. Q1: Find the prob that 1). The officer will miss the location in a given half-hour period. 2). The location will be visited twice in a given half-hour period. 3). The location will be visited at least twice in a given half-hour period. Q2: 1). Find the expectation and variance of the previous problems. 2). what does the expectation and variance tell us? Eg3.20: A type of trees has seedling randomly dispersed in a large area with mean density of seedling around five per square yard. If a forester randomly locates ten one square yard sampling regions in the area, find the prob that none of the regions will contain seedlings.

3.8 The Poisson dist. Poisson dist in R: dpois(x, lambda); ppois (x, lambda); qpois (per, lambda); rpois (N, lambda). Eg: Let X be a random variable following Pois(5). With R, find the following items: Pr(X=1); Pr(X<=10); Pr(X>6); Find x, such that Pr(X<=x)=0.8666 Find x, such that Pr(X>x)=0.735 Pr(X=1); lambda=5; dpois(x=1, lambda) Pr(X<=10); ppois (10, lambda); Pr(X>6); 1- ppois (6, lambda); Find x, such that Pr(X<=x)=0.8666 qpois (0.8666, lambda) Find x, such that Pr(X>x)=0.735 qpois (1-0.735, lambda) PHP: 3.122, 127,130,131,139,(you may use R to verify your answer )

Chap 10.6: Estimating Population Density and Size from Stocked Quadrat Samples Consider a sample of n quadrats, with equal quadrat area a, and population area A. Let y denote # of sampled quadrats that are not stocked. Under assumption of randomness of elements, proportion of unstocked quadrats in population is approximately exp{- (lamda)*a}. Then, (y/n) is an estimator of exp{-(lamda)*a}. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 8

Chap 10.6: Estimating Population Density and Size from Stocked Quadrat Samples Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 8

Eg 10.7, page 324 Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Ex10.15, page 330 Ex10.14: Air samples of 100 cubic centimeters each are taken periodically from an industrial section of a city. The density of a certain type of harmful particle is the parameter of interest. Suppose 15 samples gave an average particle count of 210 per sample. Estimate the particle density, per cubic centimeter, and place a bound on the error of estimation. EX 10.15: Suppose in the air sampling in Exercise 10.14 detecting the presence or absence of particles is easy but counting the particles is difficult. Among 500 such samples, 410 showed the particles to be present. Estimate the particle density and place a bound on the error of estimation. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Ex10.17, page 330 Q: Would you recommend use of the stocked-quadrat method for counting cars, as in Exercise 10.16? Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Ex10.18, page 330 Discuss the problem of estimating highly mobile animal populations by using quadrat sampling. Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10

Ex10.20, page 330 Outline how you can estimate the number of cars in a city during the working day. Compare four different methods for making this estimate. Which of the four do you think will work best? Why? Elementary Survey Sampling, 7E, Scheaffer, Mendenhall, Ott and Gerow Chapter 10