Normal probability between two values
For an interval, subtract the two Φ values; do not add them.
Worked example
X ~ N(μ = 75, σ = 8). What is P(69.96 ≤ X ≤ 82.68)?
Worked solution
- 1.z₁ = -0.63, z₂ = 0.96
- 2.P = Φ(0.96) − Φ(-0.63) = 0.8315 − 0.2643 = 0.5671
Typical mistake: Adding the two Φ values instead of subtracting.
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task 1 of 3Normal distribution
Task
X ~ N(μ = 75, σ = 8). What is P(61.48 ≤ X ≤ 88.92)?
P (4 decimal places)
Comma or point; fractions like 1/6 work too.
Formulas for this topic
- z = (x − μ) / σ
- P(X ≤ a) = Φ(z)
- P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
- Φ(−z) = 1 − Φ(z); quantile: x = μ + z · σ
Typical mistake: Adding the two Φ values instead of subtracting.
Frequently asked questions
What is the typical mistake in "Normal probability between two values"?
Adding the two Φ values instead of subtracting.
How do I read negative z values?
Tables usually list positive z only: Φ(−z) = 1 − Φ(z).