2 interactive labs · AS & A Level Mathematics 9709
Standardising a Normal Variable
5.5 The normal distribution · Shade a normal probability, standardise with Z = (X − μ) ÷ σ and read Φ(z) from the table, then work back from a probability to x₁, σ or μ.
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About this lab
Shade a normal probability, standardise with Z = (X − μ) ÷ σ and read Φ(z) from the table, then work back from a probability to x₁, σ or μ.
You will be able to
Calculate P(X > x₁), P(X < x₁) and P(x₁ < X < x₂) for X ~ N(μ, σ²) by standardising, reading Φ(z) to 4 d.p. from the table and using Φ(−z) = 1 − Φ(z), with the answer to 3 s.f.
Explain why the shaded area under N(μ, σ²) equals the area under the standard normal curve after standardising, and predict whether a probability is less or more than ½.
Find x₁, σ or μ from a given probability using the critical values table and the relationship x₁ = μ + zσ.
Binomial Bars and the Normal Curve
5.5 The normal distribution · Lay N(np, npq) over the bars of B(n, p), predict where to cut for ≤, <, ≥, > or =, then compare with the exact binomial probability.
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About this lab
Lay N(np, npq) over the bars of B(n, p), predict where to cut for ≤, <, ≥, > or =, then compare with the exact binomial probability.
You will be able to
State the conditions np > 5 and nq > 5 for approximating B(n, p) by N(np, npq), and recognise from the bars and the curve when they fail.
Apply the continuity correction for P(X ≤ k), P(X < k), P(X ≥ k), P(X > k) and P(X = k), moving the boundary by ½ in the correct direction.
Calculate the approximation by standardising with σ = √(npq), rounding z to 2 d.p. and reading Φ(z) from the table to 4 d.p.
Compare the approximation with the exact binomial probability, and explain why the corrected boundary is usually closer than no correction or the wrong direction (small n or a skewed p can reverse this).
