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# week3class - The Normal Distribution Normalor are a family...

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The Normal Distribution Normal—or Gaussian—distributions are a family of symmetrical, bell – shaped density curves defined by a mean μ ( mu ) and a standard deviation σ ( sigma ): N (μ, σ). Pictorially speaking, a Normal Distribution is a distribution that has a symmetric, unimodal and bell-shaped density curve. A family of density curves: 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

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All normal curves N(μ, σ) share the same properties: 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

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mean µ = 64.5 standard deviation σ = 2.5 N ( µ , σ ) = N (64.5, 2.5) Example: Women’s Heights % of young women between 62 and 67? % of young women lower than 62 or taller than 67? % between 59.5 and 62? % taller than 68.25?

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The standard normal distribution: Because all Normal distributions share the same properties, we can standardize our data to transform any Normal curve N ( μ, σ ) into the standard Normal curve N (0,1). Standardizing and Z-scores: an observation x comes from a distribution with mean μ and standard deviation σ The standardized value of x is defined as which is also called a z-score .
Mean and S.D. of the distribution of z? Example: Women’s Heights The heights of young women are approximately normal with mean = 64.5 inches and standard deviation = 2.5 inches. In our class, there is a female student who is 68.25 inches tall, what is her z- score?

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The Z-table: See handout The Z-table gives the area under the standard Normal curve to the left of any z- value.
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