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6.2 Normal Distribution http://delfe.tumblr.com/ 1 6.2/6.5: The Normal Distribution The Normal Approximation to the Binomial Distribution - Goals • • • • • Be able to sketch the normal distribution. Be able to standardize a value. Be able to use the Z-table. Be able to calculate probabilities. Be able to calculate percentiles (Inverse calculations). • Determine when you can use the normal approximation to the binomial and perform calculations using this approximation. 2 Normal Distribution 𝑓 𝑥 = 1 (𝑥−𝜇)2 − 𝑒 2𝜎2 𝜎 2𝜋 where -∞ < < ∞, σ > 0 X ~ N(,σ2) 3 Shapes of Normal Density Curve http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref /process_simulations_sensitivity_analysis_and_error_analysis_modeling /distributions_for_assigning_random_values.htm 4 Graph of Normal Distribution 5 Standard Normal Distribution (z curve) 𝑓 𝑥 = 1 (𝑥−𝜇)2 − 𝑒 2𝜎2 𝜎 2𝜋 where -∞ < < ∞, σ > 0 𝑓 𝑧 = 1 2𝜋 μ = 0, 2 = 1 𝑧2 − 𝑒 2 6 Cumulative z curve area 7 Z-table 8 Using the Z table area right of z = area between z1 and z2 = 1 area left of z area left of z1 – area left of z2 9 Procedure for Normal Distribution Problems 1. Sketch the situation and shade the area to be found. Optional but extremely useful. 2. Standardize X to state the problem in terms of Z. 3. Use Table III to find the area to the left of z. 4. Calculate the final answer. 5. Write your conclusion in the context of the problem. 𝑋−𝜇 𝑍= 𝜎 10 Percentiles 𝑋−𝜇 𝑍= 𝜎 11 Symmetrically Located Areas 12 Why Approximate the Binomial Distribution? 1. Intervals 2. Computation (n large) 3. Inference 13 Difficulties with the Normal Approximation to the Binomial 1. Skewedness of the Binomial Distribution. 2. The Binomial Distribution is discrete. 14 Continuity Correction http://wiki.axlesoft.com/index.php?title=Continuity_correction 15 Continuity Correction Actual Value P(X = a) P(a < X) P(a ≤ X) P(X < b) P(X ≤ b) Approximate Value P(a – 0.5 < X < a +0.5) P(a + 0.5 < X) P(a – 0.5 < X) P(X < b – 0.5) P(X < b + 0.5) 16