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Normal distribution
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A Probability & Statistics set, untimed
Named traps
9 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- At-least-one without complementDoes not use 1 minus P(none) for an at-least-one / minimum-sample-size problem.→
- Axis of symmetry misplacedMisidentifies the mean / interval-midpoint as the axis of symmetry (e.g. making P(x≤X≤x+2) even).→
- Binomial-approx parameters wrongComputes np or np(1-p) wrong, or uses one-tailed vs two-tailed region incorrectly, in a normal approximation.→
- Count = probability x N forgottenForgets to multiply the interval probability by the population size to estimate a count.→
- Empirical-rule band mismatchedApplies the 68-95-99.7 percentages to the wrong σ interval.→
- Mean and SD roles swappedMixes up the location (mean) and spread (standard deviation) when interpreting statements or comparing two curves.→
- Φ symmetry identity errorMisstates Φ(x)+Φ(-x)=1 or mis-converts P(|X|<x) into Φ terms.→
- Symmetry about mean not usedFails to use symmetry about the mean when finding a complementary interval probability.→
- Wrong tail / region usedUses the wrong tail or symmetric piece of the normal probability.→
Key formulas
15- Classical probabilityP(A) = n(A)/n(S), 0 ≤ P(A) ≤ 1P(certain)=1, P(impossible)=0
- Combinations C(n,k)=n!/(k!(n−k)!)permutations A(n,k)=n!/(n−k)!
- Addition ruleP(A∪B) = P(A) + P(B) − P(A∩B)
- Complement: P(Ac) = 1 − P(A)at-least-one: P(≥1) = 1 − P(none)
- Conditional: P(A|B) = P(A∩B)/P(B)Multiplication: P(A∩B)=P(A)P(B|A)Independence: P(A∩B)=P(A)P(B)
- Meanx̄ = (Σ xi)/nweighted/combined mean weights by group sizes
- Variances2 = (1/n) Σ (xi − x̄)2standard deviation s = √s2range = xmax − xmin
- Linear transformE(aX+b)=aE(X)+bD(aX+b)=a2 D(X)
- Pooled variance of two groupsweight within-group variances by size AND add between-group (mean-shift) term
- Normal PDFf(x) = 1/(σ√2π) · e−(x−μ2/(2σ2))curve symmetric about μtotal area = 1
- Empirical rule≈68% within μ±σ, ≈95% within μ±2σ, ≈99.7% within μ±3σ
- Standard normalZ = (x−μ)/σΦ(x)+Φ(−x)=1
- Estimated count in an interval = P(interval) × population size
- Least-squares slopeb̂ = (Σ xi yi − n x̄ ȳ)/(Σ xi2 − n x̄2)intercept â = ȳ − b̂ x̄
- Normal approximation to binomial (De Moivre-Laplace)X~B(n,p) ≈ N(np, np(1−p))
Shaky underneath?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
FoundationsStatistics and probability