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**Chapter 2: The Normal Distributions**

Density Curves Normal Distribution Standard Distribution

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**Density Curve Mathematical model for a distribution**

Always on or above the horizontal axis Has an area of exactly 1 There are normal, skewed left, and skewed right curves To find a value, find the area under the curve Quartile 1=1/4 of the area under the curve Quartile 3=3/4 of the area under the curve

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**Median and Mean of Density Curves**

Median—equal-areas point: divides the area under the curve in half Mean—the balance point: where the curve would balance if made solid Symbol: Mean and median are equal on a symmetric curve

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Normal Curve

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Skewed Right Curve

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Skewed Left Curve

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Normal Curve Must be: Symmetric mean=median Single Mound Bell Shaped

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**Normal Distribution Mean Standard Deviation**

Can be written—N(mean, standard deviation) Changing the Standard Deviation changes the spread of the curve Changing the mean without changing the standard deviation moves the graph horizontally on the x-axis Larger Standard Deviation=curve more spread out Inflection Point-points where the curvature change takes place Located a from each side of the mean

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**The 68-95-99.7 Rule 68% of all observations fall within of**

To find an observation’s percentile, use the Z-Score

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Graph of Rule

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Example The distribution of heights of adult American men is approximately normal with mean of 69 inches and a standard deviation of 2.5 inches. Between what heights do the middle 68% of men fall? 69+2.5=71.5 69-2.5=66.5 From 66.5 to 71.5

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**Z-Score To find what percentage a certain individual/value is**

Tells us how many standard deviation we are from the mean Formula- Can use the table or the calculator Calculator normalcdf(lower bound, upper bound, mean, standard deviation)

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Z-Score Table

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**Assessing Normality Method 1—Frequency Histogram or Stem Plot**

Approximately bell-shaped Symmetric about the mean Check to see if rule works Method 2--Normal Probability Plot Interpret the graph—if the plotted points appear to be close to a line, then it is a normal distribution

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The End

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