ZIMSEC A Level · Statistics · Form 5, Form 6
Properties
X ~ N(μ, σ²) is continuous, symmetrical and bell-shaped about the mean μ. About 68% of values lie within 1σ of the mean, 95% within 2σ (1.96σ exactly) and 99.7% within 3σ.
Standardising
Z = (X − μ)/σ ~ N(0, 1). Use tables of Φ(z) = P(Z < z).
Worked example: Masses of maize bags are N(50, 4) kg (σ = 2). Find P(a bag weighs less than 53 kg).
z = (53 − 50)/2 = 1.5 → P = Φ(1.5) = 0.9332
Worked example: Find P(48 < X < 53).
z-values −1 and 1.5 → Φ(1.5) − Φ(−1) = 0.9332 − 0.1587 = 0.7745
Inverse problems
Given a probability, find z from tables, then x = μ + zσ, or solve simultaneous equations for unknown μ and σ.
Worked example: The top 10% of marks are above what value if marks are N(60, 100)?
z = 1.282 → x = 60 + 1.282 × 10 = 72.8
Normal approximations
- To B(n, p) when np > 5 and n(1 − p) > 5: N(np, np(1 − p)) with a continuity correction.
- To Po(λ) when λ is large: N(λ, λ).
Check yourself
- X ~ N(20, 9). Find P(X > 23).
- What percentage of a normal distribution lies within 1.96 standard deviations of the mean?
- Write the continuity correction for P(X ≤ 15) when approximating a binomial by a normal.
Show answers
- z = 1 → 1 − 0.8413 = 0.1587
- 95%
- P(Y < 15.5)
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