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Shape of Normal Curves

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68%-95%-99.7% Rule

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Areas under Normal Curve

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Areas under Normal Curve(cont)

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Example: Normal Distribution The brain weights of adult Swedish males are approximately normally distributed with mean μ = 1,400 g and standard deviation = 100 g. (No real life population follows a normal distribution exactly!) a) What is the probability that an adult Swedish male has a brain weight of less then 1,500 g? b) What is the probability that an adult Swedish male has a brain weight between 1,475 g and 1,600 g?

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Example: Normal Distribution (cont) μ = 1,400 g and = 100 g a) What is the probability that an adult Swedish male has a brain weight of less then 1,500 g?

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Example: Normal Distribution (cont) μ = 1,400 g and = 100 g b) What is the probability that an adult Swedish male has a brain weight between 1,475 g and 1,600 g?

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Area under the normal curve above

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Example: Normal Distribution The brain weights of adult Swedish males are approximately normally distributed with mean μ = 1,400 g and standard deviation = 100 g. (No real life population follows a normal distribution exactly!) c) What is the 55 th percentile for the distribution of brain weights?

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Example (ExDispersion.sas) Determine the percentage of data points within 1 SD? 2 SD? 721124161210136 1218151636911

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Example: Normality (ExNormal.sas) 721124161210136 1218151636911

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Example: QQPlots – Normal (ExQQplot.sas)

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Example: QQPlots – Right Skewed

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Example: QQPlots – Left Skewed

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Example: QQPlots – Long Tail

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Example: QQPlots – Tails?

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Example 4.4.5: Nonnormal Data

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Interpretation of Shapiro-Wilk Test P-ValueInterpretation < 0.001Very strong evidence for nonnormality < 0.01Strong evidence for nonnormality < 0.05Moderate evidence for nonnormality < 0.10Mild or weak evidence for nonnormality 0.10 No compelling evidence for nonnormality

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Objective Measure: SAS Tests for Normality TestStatisticp Value Shapiro-WilkW0.98762Pr < W0.8757 Kolmogorov-SmirnovD0.092489Pr > D>0.1500 Cramer-von MisesW-Sq0.042289Pr > W-Sq>0.2500 Anderson-DarlingA-Sq0.233462Pr > A-Sq>0.2500

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Objective Measure: SAS Tests for Normality TestStatisticp Value NormalW0.98762Pr < W0.8757 Right SkewedW0.949844Pr > W0.4226 Left SkewedW0.925624Pr > W0.0479 Long TailedW0.927118Pr > W0.0043 Short TailedW0.949227Pr > W0.0317

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Example: QQPlots x

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Example 4.10: Continuity Correction Table 4.1 shows the distribution of litter size for a population of female mice with population mean 7.8 and SD 2.3. x

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Example 4.10: Continuity Correction(cont) Table 4.1 shows the distribution of litter size for a population of female mice with population mean 7.8 and SD 2.3. x

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