Lecture_Statistics_Spring_2013b

Probability distributions confidence intervals

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Unformatted text preview:  expȺ− Ⱥ dx = p(X ≤ A) ∫ 2 πσ 2σ 2 Ⱥ Ⱥ −∞ Ⱥ Ⱥ A 1 III. Probability Distributions & Confidence Intervals Standard Normal Distribution p( z ) = where Ⱥ z 2 Ⱥ expȺ− Ⱥ 2π Ⱥ 2 Ⱥ Ⱥ Ⱥ 1 ( X − µ) z= σ * Similarly, the integral from negative infinity to X = A is the probability of observing a value less than or equal to A: Ⱥ z 2 Ⱥ expȺ− Ⱥ dx = p(z ≤ A) ∫ Ⱥ 2 Ⱥ − ∞ 2π Ⱥ Ⱥ A 1 See Table 1 III. Probability Distributions & Confidence Intervals Confidence Intervals N observations, if true population variance is known, 100(1-α)% confidence interval is given as: x ± zα / 2 σ N Typical values for zα/2 are 1.645, 1.96, and 2.575 for 90, 95, and 99% confidence intervals, respectively. III. Probability Distributions & Confidence Intervals If the population variance is not known, The standard normal Gaussian probability distribution function is replaced by Student's t-distribution Student’s t-distribution υ +1 ȹ υ + 1 ȹ − Γȹ ȹ Ⱥ 2 ȹ 2 Ⱥ 1 + t Ⱥ 2 p( t ) = Ⱥ Ⱥ υ Ⱥ ȹ υ ȹ Ⱥ Ⱥ υπΓȹ ȹ Ⱥ ȹ 2 Ⱥ Γ(i): gamma functions The degrees of freedom: ν = N – 1 t is defined as: (X − x ) t= υ +1 ȹ υ + 1 ȹ − Γȹ ȹ Ⱥ A t 2 Ⱥ 2 2 Ⱥ ȹ dt = p( t ≤ A) Ⱥ1 + Ⱥ ∫ υ Ⱥ − ∞ υπΓȹ υ ȹ Ⱥ Ⱥ Ⱥ ȹ ȹ ȹ 2 Ⱥ s See Table 2 III. Probability Distributions & Confidence Intervals Figure shows t...
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