Master the standard normal curve — z-scores, probabilities, and symmetry. A z-table is provided below. Each question builds on the last.
\(z = \dfrac{x - \mu}{\sigma}\)
x = value, μ = mean, σ = std dev
CDF gives the area to the LEFT of x.
Always left-tail by default.
P(X > x) = 1 − Φ(z)
Flip it! Total area = 1.
Negative z → mirror the positive side.
Φ(−1.5) = 1 − Φ(1.5)
P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
Big minus small.
±1σ → 68%
±2σ → 95%
±3σ → 99.7%
Each cell shows Φ(z) = P(Z ≤ z), the cumulative probability from −∞ to z.