A population geneticist is studying the genes found at two different locations on the genome. He estimates the proportion p̂₁ of organisms who have gene A at the first locus to be p̂₁ = 0.42, with uncertainty σ₁ = 0.049. He estimates the proportion of organisms that have gene B at a second locus to be p̂₂ = 0.23, with uncertainty σ₂ = 0.043. Under assumptions usually made in population genetics (Hardy–Weinberg equilibrium), p̂₁ and p̂₂ are independent and normally distributed, and the proportion p of organisms that have both genes A and B is estimated with p̂ = p̂₁ p̂₂.

a. Compute p̂ and use propagation of error to estimate its uncertainty.

b. Assuming p̂ to be normally distributed, find the P-value for testing H₀ : p ≥ 0.10.

c. Generate an appropriate simulated sample of values p̂^∗. Is it reasonable to assume that p̂ is normally distributed?