For X ~ N(120, 15²), compute P(one invoice > 132), P(mean of 25 > 132) and P(total of 25 > 3150) with the three SDs σ, σ/√n and √n·σ.

Answer

import math def phi(z): return 0.5 * (1 + math.erf(z / math.sqrt(2))) mu, sigma, n = 120, 15, 25 one = 1 - phi((132 - mu) / sigma) mean = 1 - phi((132 - mu) / (sigma / math.sqrt(n))) total = 1 - phi((3150 - n * mu) / (math.sqrt(n) * sigma)) print(f"individual={one:.4f} mean25={mean:.4f} total25={total:.4f}")

IS1 §7.3 (the three-way rule and the sum), CC BY 4.0 — shape only; H-W7 (single value vs average) — shape only; original dataset