Stratified Cruise Design With a Precision Target
A consulting forester is designing a timber cruise for a 120-hectare tract before a sale. The tract has two clearly separable strata: 70 hectares of upland pine and 50 hectares of bottomland hardwood. A pilot cruise of 20 randomly located plots per stratum produced these preliminary estimates: upland pine mean volume 185 m³/ha with standard deviation 55 m³/ha; bottomland hardwood mean 120 m³/ha with standard deviation 70 m³/ha. The landowner wants the whole-tract mean volume estimated within ±10% at 95% confidence. Access is difficult in the bottomland after rain, and the sale deadline is tight. The forester must decide sample size, allocation, and field quality control. A previous cruise on the same property used 40 plots and produced a confidence interval of ±18%, which the landowner rejected as too imprecise. The client also wants volume by product class, not just total volume.
1. Calculate the approximate number of plots needed for the whole-tract precision target and allocate them between strata. State your assumptions and show the reasoning.
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First estimate the whole-tract mean from the pilot data: (70 × 185 + 50 × 120) / 120 = 159.6 m³/ha. A ±10% target is therefore an absolute error of about 15.96 m³/ha at 95% confidence, using t ≈ 1.96 for a preliminary calculation.
For stratified sampling with optimal allocation, the required number of plots is approximately n = t² (Σ N_h s_h)² / (N² E²). Here Σ N_h s_h = 70 × 55 + 50 × 70 = 3,850 + 3,500 = 7,350. Thus n = 1.96² × 7,350² / (120² × 15.96²) ≈ 56.6, so round up to 57 plots. Allocate in proportion to N_h s_h: pine gets 57 × 3,850 / 7,350 ≈ 30 plots; hardwood gets 57 × 3,500 / 7,350 ≈ 27 plots. Because the pilot standard deviations are uncertain, add a small buffer, for example 10%, and use the actual t value if the final sample is below 30 per stratum. A finite population correction would reduce the number only if the number of possible plots is known; that information is not given.
2. The client also wants volume by product class. How does that change the design, and what would you tell the client about precision?
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A design that meets ±10% for total volume does not automatically meet ±10% for sawtimber, pulpwood, or species groups. Product-class estimates depend on the proportion of trees in each class, defect, and diameter distribution, so their variance can be much larger than total volume variance. I would record species, DBH, total height, merchantable height, defect, and crown class on every plot, and define product classes before fieldwork.
I would then check whether the product-class proportions are adequate. If, for example, sawtimber is a small fraction of volume, the 57-plot design may produce a wide interval around that fraction even though total volume is precise. Options are to increase sample size, allocate extra plots to the stratum with more sawtimber, or use a two-phase design in which a larger number of plots is measured for species and DBH and a subsample is measured for height and defect. I would tell the client plainly that total volume and product-class volume carry different confidence intervals, and I would not promise ±10% for every product class unless the design is checked against those separate objectives.
3. What field quality-control procedures would you use, including rules for borderline plots and difficult access?
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Quality control should be built into the cruise, not added at the end. Train all cruisers on the same diameter, height, defect, and product-class rules; calibrate height instruments; and use a written plot-location rule. A common rule is to include a plot when its center falls inside the stratum boundary, with a consistent treatment of plots that straddle the boundary. Record GPS coordinates and slope-corrected horizontal distances.
Rem easure about 10% of plots independently, preferably by a different cruiser, and compare species, DBH, height, defect, and volume. Investigate any systematic difference before continuing. For the difficult bottomland access, do not substitute easily reached plots for random plots; that biases the sample. Use access corridors, wait for drier conditions, or pre-select alternate random plots under a documented rule. If access truly prevents sampling some plots, report them as non-response and adjust the estimator or expand the sample rather than quietly shifting plot locations.