The Applied units cover Statistics (S1/S2: data, probability, distributions, hypothesis testing) and Mechanics (M1/M2: kinematics, forces, moments, and dynamics) — applying Pure Maths to real-world problems.
Data: histograms (frequency density), cumulative frequency, box plots. Mean, median, mode, range, IQR, standard deviation. Probability: addition rule P(A∪B) = P(A)+P(B)−P(A∩B). Conditional P(A|B) = P(A∩B)/P(B). Tree diagrams, Venn diagrams. Binomial distribution B(n,p): P(X=r) = ⁿCᵣ pʳ(1−p)ⁿ⁻ʳ. Mean np, variance np(1−p). Normal distribution N(μ,σ²): standardise Z = (X−μ)/σ, use standard normal table. Hypothesis testing: H₀ (null) vs H₁ (alternative). Test statistic, significance level, critical region, p-value. Conclusion in context.
Kinematics: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u+v)t. Displacement-time and velocity-time graphs. Variable acceleration: v = ds/dt, a = dv/dt; s = ∫v dt. Forces: weight W = mg, normal reaction, friction F ≤ μR. Newton\'s 2nd law: F = ma (resolve in direction of motion). Connected particles: systems (pulleys, towing). Moments: moment = force × perpendicular distance from pivot. Equilibrium: sum of clockwise moments = sum of anticlockwise moments, and resultant force = 0. Centre of mass for composite shapes.
Binomial B(n,p): for counting successes in a fixed number of independent trials, each with the same probability of success. Conditions: fixed n, independent, two outcomes, constant p. Discrete values: X = 0,1,2,...,n. Normal N(μ,σ²): for continuous measurements that cluster symmetrically around a mean — heights, weights, errors. Bell-shaped, extends to ±∞. Connection: if n is large and p isn\'t near 0 or 1, B(n,p) ≈ N(np, np(1−p)). Rule of thumb: np > 5 and n(1−p) > 5. When approximating, apply a continuity correction: e.g., P(X ≥ 10) for binomial becomes P(X > 9.5) for normal. In exams, the question context tells you which: "number of successes" → binomial; "measurements" → normal.
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