# Shape of Normal Curves. 68%-95%-99.7% Rule Areas under Normal Curve.

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Shape of Normal Curves

68%-95%-99.7% Rule

Areas under Normal Curve

Areas under Normal Curve(cont)

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?

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?

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?

Area under the normal curve above 

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?

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

Example: Normality (ExNormal.sas) 721124161210136 1218151636911

Example: QQPlots – Normal (ExQQplot.sas)

Example: QQPlots – Right Skewed

Example: QQPlots – Left Skewed

Example: QQPlots – Long Tail

Example: QQPlots – Tails?

Example 4.4.5: Nonnormal Data

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

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

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

Example: QQPlots x

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

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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