Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — a probability between two values
X∼N(50,64). Find P(45<X<60).
- 1The variance is 64, so σ=8.Take the square root — the second parameter is the variance.
- 2Standardise the upper bound: z=860−50=1.25
- 3Standardise the lower bound: z=845−50=−0.625
- 4Φ(1.25)=0.8944 and Φ(−0.625)=1−Φ(0.625)=0.2660Symmetry handles the negative z.
- 5P=0.8944−0.2660=0.6284A sketch confirms the region spans most of the middle of the curve ✓
Example 2 — working backwards from a probability
X∼N(50,64). Find k such that P(X<k)=0.90.
- 1Find the z value with 0.90 of the area to its left.Read the table in reverse.
- 2Φ(z)=0.90 gives z=1.2816.Positive, since 0.90 exceeds half.
- 3Reverse the standardisation: X=μ+zσ
- 4=50+8(1.2816)
- 5k=60.3 (3 s.f.)Above the mean, as an upper 90% boundary must be ✓
Example 3 — a normal approximation with continuity correction
X∼B(100,0.4). Estimate P(X≤45).
- 1Check the conditions: np=40>5 and nq=60>5 ✓State this — it justifies the method.
- 2μ=np=40 and σ=npq=24=4.899
- 3Continuity correction: P(X≤45) becomes P(Y<45.5).45 is included, so the boundary moves outwards.
- 4z=4.89945.5−40=1.123
- 5Φ(1.123)=0.869
- 6The exact binomial value is 0.8689 ✓Without the correction the estimate would be 0.846 — over 2 points out.