The Normal Distribution

• X ~ N(μ, σ²) is a continuous, symmetrical, bell-shaped distribution — note the second parameter is the variance.
• Standardise: Z = (X − μ)/σ, with Z ~ N(0, 1); show this working in full.
• Use the table for Φ(z) and symmetry: Φ(−z) = 1 − Φ(z).
• Given a probability, use the inverse table (critical values) to find z, then form equations in μ and σ.
• Normal approximation to B(n, p) when np > 5 and nq > 5: use N(np, npq) with a continuity correction.

Standardise, sketch, look up

Sketch the curve, shade the region you want, standardise the boundary values and look up Φ(z) = P(Z < z). For areas to the right use 1 − Φ(z); for negative z use symmetry. For a continuous variable, P(X < a) = P(X ≤ a).

Probability → z → equation

If a probability is given, read the z-value from the inverse table (watch the sign: a value below the mean has negative z). Each probability gives one equation (x − μ)/σ = z; with two unknowns you need two equations, solved simultaneously.

Continuity correction

If X ~ B(n, p) with n large enough that np > 5 and nq > 5 (q = 1 − p), X is approximately N(np, npq). The binomial is discrete, so each integer r is represented by the interval r − 0.5 to r + 0.5: this is the continuity correction.