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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 Medianequal-areas point: divides the area under the curve in half Meanthe 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 writtenN(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 Rule 68% of all observations fall within of 95% of all observations fall within of 99.7% of all observations fall within of To find an observations 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? = =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 1Frequency Histogram or Stem Plot – Approximately bell-shaped – Symmetric about the mean – Check to see if rule works Method 2--Normal Probability Plot – Interpret the graphif the plotted points appear to be close to a line, then it is a normal distribution

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