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Shape of Normal Curves. Shape of Normal Curves. 68%-95%-99.7% Rule. Areas under Normal Curve. Areas under Normal Curve(cont). Example: Normal Distribution. - PowerPoint PPT Presentation

### Transcript of Shape of Normal Curves

Proposition 1.1 De Moargans Laws

Shape of Normal Curves

1Shape of Normal Curves

268%-95%-99.7% Rule

3Areas under Normal Curve

4Areas under Normal Curve(cont)

5Example: Normal DistributionThe brain weights of adult Swedish males are approximately normally distributed with mean = 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?6Example: Normal Distribution (cont) = 1,400 g and = 100 ga) What is the probability that an adult Swedish male has a brain weight of less then 1,500 g?

7Example: Normal Distribution (cont) = 1,400 g and = 100 gb) What is the probability that an adult Swedish male has a brain weight between 1,475 g and 1,600 g?

8Area under the normal curve above

Example: Normal DistributionThe brain weights of adult Swedish males are approximately normally distributed with mean = 1,400 g and standard deviation = 100 g. (No real life population follows a normal distribution exactly!)c) What is the 55th percentile for the distribution of brain weights? 10

Example (ExDispersion.sas)Determine the percentage of data points within 1 SD? 2 SD? 7211241612101361313131218151636911Example: Normality (ExNormal.sas)7211241612101361313131218151636911

Example: QQPlots Normal (ExQQplot.sas)

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Example: QQPlots Right Skewed

14Example: QQPlots Left Skewed

15Example: QQPlots Long Tail

16Example: QQPlots Tails?

17Example 4.4.5: Nonnormal Data

Interpretation of Shapiro-Wilk TestP-ValueInterpretation< 0.001Very strong evidence for nonnormality< 0.01Strong evidence for nonnormality< 0.05Moderate evidence for nonnormality< 0.10Mild or weak evidence for nonnormality 0.10No compelling evidence for nonnormalityObjective Measure: SASTests for NormalityTestStatisticp ValueShapiro-WilkW0.98762Pr < W0.8757Kolmogorov-SmirnovD0.092489Pr > D>0.1500Cramer-von MisesW-Sq0.042289Pr > W-Sq>0.2500Anderson-DarlingA-Sq0.233462Pr > A-Sq>0.2500

Objective Measure: SASTests for NormalityTestStatisticp ValueNormalW0.98762Pr < W0.8757Right SkewedW0.949844Pr > W0.4226Left SkewedW0.925624Pr > W0.0479Long TailedW0.927118Pr > W0.0043Short TailedW0.949227Pr > W0.0317Example: QQPlots

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

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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.

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