Normal: 31% under 45, 8% over 64

Let X ~ N(μ, σ²). Given: P(X < 45) = 0.31 and P(X > 64) = 0.08.

Z-values

• P(X < 45) = 0.31 lies below the mean, so the area between 45 and μ is 0.50 − 0.31 = 0.19. Given Z₀.₁₉ = 0.5, the z-value is −0.5.
• P(X > 64) = 0.08, so the area between μ and 64 is 0.50 − 0.08 = 0.42. Given Z₀.₄₂ = 1.4, the z-value is +1.4.

Equations

(45 − μ)/σ = −0.5 → μ − 0.5σ = 45 ... (1) (64 − μ)/σ = 1.4 → μ + 1.4σ = 64 ... (2)

Subtract (1) from (2): 1.9σ = 19 → σ = 10

From (1): μ = 45 + 0.5 × 10 = 50

(Check: (64 − 50)/10 = 1.4.)

Answer: mean = 50, standard deviation = 10