Question

1. The archaeological site of Tara is more than 4,000 years old. Tradition states that Tara was the seat of the high kings of Ireland. Because of it's archaeological importance, Tara has received extensive study (Reference: Tara: An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Suppose an archaeologist wants to estimate the density of ferromagnetic artifacts in the Tara region. For this purpose, a random sample of 55 plots, each of size 100 square meters, is used. The number of ferromagnetic artifacts for each plot is determined. ( /3 points)

Answer 1

I don't see any observed counts in your prompt, so I'll state the parameter, the natural estimator, and the formulas you would use (and how to get a confidence interval).

Setup
- n = 55 independent random plots, each of area A = 100 m².
- Yi = number of ferromagnetic artifacts in plot i (i = 1,…,55).
- Let μ = E[Yi] (mean count per 100 m²). The quantity of interest (density) per square meter is
λ = μ / A.

Point estimator
- Sample mean count per plot: Ȳ = (1/n) Σ Yi.
- Estimate of density per m²:
λ̂ = Ȳ / A.

Estimated standard error
- Let s² = sample variance of the Yi. Then
Var(λ̂) ≈ s² / (n A²),
so
SE(λ̂) = s / (√n A).
- If a Poisson model is plausible (Var(Yi) ≈ μ), use s² ≈ Ȳ to get
SE(λ̂) ≈ √(Ȳ / (n A²)).

Approximate 95% confidence interval
- Using normal approximation (n = 55 is large enough):
λ̂ ± z0.975 · SE(λ̂) (z0.975 ≈ 1.96).
- Alternatively use t0.975,n−1 if you prefer the t-distribution.

Notes
- To report density per hectare multiply λ̂ (per m²) by 10,000.
- If counts are rare and many zeros occur, consider modelling with a Poisson or negative-binomial GLM and base inference on that model.

If you give the observed counts (or their sample mean and sample variance), I can compute λ̂, its SE, and a 95% CI numerically.