Sampling theory
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Stratified random sampling involves grouping a population into homogeneous strata and then randomly sampling within each strata. This allows for a more precise estimate of the overall population than simple random sampling. The document provides an example comparing simple random sampling to stratified random sampling when cruising a 500 acre forest tract divided into three cover types. Stratified random sampling resulted in a more precise cruise with a standard error of 19.4 cfv/a compared to 38.9 cfv/a for simple random sampling. An optimum sample size of 15-16 plots was calculated to estimate volume within 10% precision, with 7 plots allocated to the first stratum, 4 to the second, and 5 to the third.
Sampling theory
- 1. LECTURES FOR WEEK 11 Stratified Random Sampling: So far, we have discussed random and systematic location of cruise plots in the context of simple random sampling. These methods work well when the area sampled is homogeneous. Foresters, however, often cruise areas with different forest cover types (i.e., stands). These different stands make the area heterogeneous in terms of cover types. In this case, stratified random sampling can be used to calculate a more precise cruise. In stratified random sampling, the units of the population (e.g., stands) are grouped together based on similar criterion (e.g., overstory tree type). Each unit (stratum) or stand is then cruised and the stratum estimates are combined to give an estimate for the entire area. We will illustrate this methodology with an example. EXAMPLE: You cruise a 500 acre tract using variable radius plots and determine the following cubic foot volumes per acre for cruise points in three timber types: I. Black Cherry Maple (SAF Cover Type 28) PLOT VOLUMES (cubic foot volume per acre) 570 640 480 560 510 590 670 600 780 700 1
- 2. II. Yellow-Poplar White Oak Northern Red Oak (SAF Cover Type 59) PLOT VOLUMES (cubic foot volume per acre) 520 710 770 840 630 760 890 810 580 860 III. White Oak Black Oak Northern Red Oak (SAF Cover Type 52) PLOT VOLUMES (cubic foot volume per acre) 200 420 210 290 350 540 180 260 320 270 Mean: First, calculate the mean cubic foot volume per acre (CFV/A) for each stratum using the simple random procedure we used earlier: X I = 6100 / 10 = 610 cfv/a X II = 7370 / 10 = 737 cfv/a X III = 3040 / 10 = 304 cfv/a The mean of the stratified sample can now be computed by: L N h Xh X ST = h =1 , N where: L = The number of strata Nh = The size of stratum h (h = 1, 2, , L) in acres. L N = The total size of the tract in acres ( N = N h ). h =1 2
- 3. If the strata sizes are: NI = 250 acres NII = 100 acres NIII = 150 acres Total Acres: N = 500 acres. Then the stratified sample mean cubic volume per acre is: 250 *610 + 100 * 737 + 150 * 304 X ST = = 543.6 cfv/a 500 If you used the simple random sampling formula to calculate the mean, you would get: 16,510 XS = = 550.3 cfv/a 30 which is close to the stratified mean. Standard Errors: Next, calculate the variance ( s 2 ) of cubic foot volume per acre (CFV/A) for each stratum using the simple random methodology we used earlier: s 2 = 8111.1 cfv/a I s 2 = 15,556.7 cfv/a II s 2 = 12,204.4 cfv/a III With these variances, calculate the stratified standard error of cubic foot volume per acre for the sample by: L sh SE ST = w 2 h , h =1 nh where nh = number of cruise plots wh = weight factor = Nh / N. 3
- 4. So, the stratified standard error for this example is: 2 2 2 250 8111.1 100 15,556.7 150 12,204.4 SE SR = + + = 19.4 cfv/a , 500 10 500 10 500 10 If you used the simple random sampling formula to calculate the standard error, you would get: SES = 38.9 cfv/a, which is greater than the stratified standard error. Thus, stratifying the sample will improve our overall cruise precision in this example (why?). 95% Confidence Interval: Calculate the 95% confidence interval for cubic foot volume per acre: X ST 賊 t0.05,n-1=29 * SEST 543.6 賊 2.045*19.4 or 543.6 賊 39.7 cfv/a Cruise Precision: Calculate the cruise precision for this stratified cruise: SE ST * t 0.05, 29 19.4 2.045 Pr ecision = = 100 7% X ST 543.6 Pr ecision (SRS) 15% Sample Size: You can calculate the number of plots needed in each stratum that are necessary to achieve a specified statistical objective. First, determine the total number of plots necessary by: 2 L t * w h * s xh 2 n= h =1 , E2 4
- 5. where: E = allowable error in absolute units all other variables defined as before. So, for our example, we want to find the number of plots needed to be 95% confident that we are within plus or minus 10% of the true cubic foot volume per acre. In absolute units, E = 543.6 cfv/a * (0.10) = 54.4 cfv/a. 2 250 100 150 22 * * 8111.1 + * 15,556.7 + * 12,204.4 n= 500 500 500 15 plots 2 54.4 For our example, we actually collected more sample points than necessary to achieve this statistical objective. Continuing with our example, we can now allocate (i.e., optimum allocation) these 15 plots to the three strata with the formula: w h * s xh nh = L *n w h =1 h * s xh So, 250 * 8111.1 nI = 500 * 15 = 45.03 * 15 7 plots 250 100 150 103.12 * 8111.1 + * 15,556.7 + * 12,204.4 500 500 500 100 * 15,556.7 n II = 500 *15 4 plots 103.12 5
- 6. 150 * 12,204.4 n III = 500 * 15 5 plots 103.12 You will notice that the total number of plots equals 16 if you add up the strata plot numbers. This number is greater than 15 because you should always round up the result when calculating sample size. You should also know that if you had an estimate of variability and you defined your strata before the cruise, you can calculate your sample size before the cruise and allocate the plots to each stratum with the same methodology. 6