A soil scientist uses spectroscopy to classify soil types across 200 plots: 30% sandy, 50% loamy, 20% clay. She samples 5 plots from each type. If she runs a chemical analysis on each, what is the minimum number of samples that must be high-clay to ensure at least 3 samples are from high-clay plots, assuming worst-case random placement? - ToelettAPP

Understanding the Context

The 200 plots break down as follows:

• Sandy soil: 30% × 200 = 60 plots
  
• Loamy soil: 50% × 200 = 100 plots
  
• Clay soil: 20% × 200 = 40 plots

From each type, 5 plots are sampled, totaling 15 samples. The study aims to determine the minimum number of high-clay samples required to ensure at least 3 confirmed high-clay results—under worst-case sampling conditions.

Understanding Worst-Case Sampling

Key Insights

Worst-case sampling assumes the most unpredictable or unfavorable distribution of high-clay plots within the sampled sites. However, the goal is not just estimation—it’s guaranteeing results. To minimize the number of high-clay samples needed for at least 3 confirmations, we must consider the maximum number of low- and medium-clay samples that could be selected before reaching the target.

Let’s summarize the counterfactual:

• Total high-clay plots: 40
  
• Total low- and medium-clay plots: 60 + 100 = 160

If samples are sampled randomly, the worst case for satisfying the 3-high-clay condition occurs when as many low- and medium-clay samples as possible fill the 5 slots per type—until only limited high-clay samples remain.

Sampling Strategy and Minimum High-Clay Samples

Since 5 samples are taken from each of the three soil types:

• Loamy (100 plots) and sandy (60 plots) have ample supply—context doesn’t limit random sampling assumption.
  
• Clay soil has only 40 plots, so only 5 samples can be drawn, but we care about distribution across types.

Final Thoughts

The key is: how many of the 15 total samples can not be from high-clay plots? Up to 10 samples could theoretically come from non-high-clay soils (5 from loamy + 5 from sandy) if high-clay plots are avoided in those slots—up to the 40 available.

To guarantee at least 3 high-clay samples, we must assume the adversary picks samples to delay reaching this threshold. That means maximizing non-high-clay selections first.

Maximum non-high-clay samples possible:
– 100 loamy + 60 sandy = 160 plots
But only 5 per soil type are sampled — so up to 5 from loamy, 5 from sandy, and 5 from clay, but to maximize non-high-clay usage, assume crew selects only loamy and sandy as much as possible.

But clay plots exist: we must account for worst-case inclusion of high-clay samples.

The weakest strategy to avoid high-clay samples picks up to 160 non-clay samples from loamy and sandy plots — but clay only has 40 plots. So maximum non-clay samples possible is limited.

However, since only 5 samples per type are taken, the maximum number of non-high-clay samples is capped by the number of loamy (100) and sandy (60), but clay plots can still be avoided only if sampling does not hit them.