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More applications of the Z-Score NORMAL Distribution The Empirical Rule

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Normal Distribution? These density curves are symmetric, single-peaked, and bell-shaped. We capitalize Normal to remind you that these curves are special. Normal distribution is described by giving its mean μ and its standard deviation σ

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Shape of the Normal curve The standard deviation σ controls the spread of a Normal curve

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The Empirical Rule rule In the Normal distribution with mean μ and standard deviation σ: Approximately 6 8% of the observations fall within σ of the mean μ. Approximately 9 5% of the observations fall within 2σ of μ. Approximately 9 9.7% of the observations fall within 3σ of μ.

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The Normal Distribution and Empirical Rule

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Example: YOUNG WOMEN’s HEIGHT The distribution of heights of young women aged 18 to 24 is approximately Normal with mean μ = 64.5 inches and standard deviation σ = 2.5 inches.

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Importance of Normal Curve scores on tests taken by many people (such as SAT exams and many psychological tests), repeated careful measurements of the same quantity, and characteristics of biological populations (such as yields of corn and lengths of animal pregnancies). even though many sets of data follow a Normal distribution, many do not. Most income distributions, for example, are skewed to the right and so are not Normal

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Standard Normal distribution Standard Normal Distribution The standard Normal distribution is the Normal distribution N (0, 1) with mean 0 and standard deviation

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Standard Normal Calculation The Standard Normal Table Table A Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z.

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Area to the LEFT Using the standard Normal table Problem: Find the proportion of observations from the standard Normal distribution that are less than illustrates the relationship between the value z = 2.22 and the area How to use the table of values

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illustrates the relationship between the value z = 2.22 and the area

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Example Area to the RIGHT Using the standard Normal table Problem: Find the proportion of observations from the standard Normal distribution that are greater than −2.15 z = −2.15 Area = Area = Area =.9842

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Practice (a) z < 2.85 (b) z > 2.85 (c) z > −1.66 (d) −1.66 < z < 2.85 (a) (b) (c) (d)

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CODY’S quiz score relative to his classmates x z = 0.99 Area =.8389 Cody ’ s actual score relative to the other students who took the same test is 84%

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