ST107 Quantitative Methods (CORRECTLY SOLVED!)

Mean & Variance of transformed distributions correct answers Mean: Affected by all operations (additions/subtraction/multiplication/division) Variance: Only affected by changes in scale (multiplication/division), must square scale changes --- i.e. if original variance is 6 and all values divided by 2, new variance = 6/2^2 Axioms of Probability correct answers 1)For any event A, 0 ≤ P(A) ≤ 1 2)For sample space S, P(S) = 1 3)If {Ai}, i = 1,..., n are mutually exclusive events then the probability of their union is the sum of their respective probabilities Conditions for Probability Distribution Functions correct answers 1) f(x) ≥ 0 for all values of x 2) ∫f(x) dx = 1 Transformation Formula for Standardisation correct answers Z = (X-µ)/σ when X ~ N(µ, σ2) --- remember to square root SD to find variance! Poisson Approximation to Normal correct answers Approximate X ∼ Bin(n, π) with Y ∼ Pois(nπ) Normal Approximation to Binomial/Poisson correct answers Approximate X ~ Bin(n, π) with Y ~ N(nπ, nπ(1-π)) Approximate X ~ Pois(λ) with Y ~ N(λ,λ) • Continuity Correction: Used when transitioning from a discrete to a continuous distribution --- either add or subtract 0.5 of a unit from each discrete x-value - If P(X = n) use P(n - 0.5 X n + 0.5) - If P(X n) use P(X n + 0.5) - If P(X ≥ n) use P(X n - 0.5) - If P (X n) use P(X n - 0.5) - If P(X ≤ n) use P(X n + 0.5) Normal Approximation of Student T Distribution: correct answers ν → ∞, tν → N(0,1) Central Limit Theorem correct answers n → ∞, x̄ ~ N(µ, σ2/n). Mean Square Error MSE correct answers Mean Square Error: Provides a mechanism for evaluating the relative performance of different estimators taking into account bias and variance Ways of Reducing Confidence Level: correct answers 1) Reduce level of confidence 2) Increase sample size P-value correct answers The probability of observing the test statistic value or a more extreme value conditional on H0 being true