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Section 6-1 Introduction to Normal Distributions

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Section 6-1 Objectives Interpret graphs of normal probability distributions Find areas under the standard normal curve

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Properties of Normal Distributions Normal distribution A continuous probability distribution for a random variable, x. The most important continuous probability distribution in statistics. The graph of a normal distribution is called the normal curve.

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Properties of Normal Distributions The mean, median, and mode are equal. The normal curve is bell-shaped and is symmetric about the mean. The total area under the normal curve is equal to 1. The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean.

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Properties of Normal Distributions Between μ – σ and μ + σ (in the center of the curve), the graph curves downward. The graph curves upward to the left of μ – σ and to the right of μ + σ. The points at which the curve changes from curving upward to curving downward are called the inflection points.

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Means and Standard Deviations A normal distribution can have any mean and any positive standard deviation. The mean gives the location of the line of symmetry. The standard deviation describes the spread of the data.

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Example: Understanding Mean and Standard Deviation Which normal curve has the greater mean?

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Example: Understanding Mean and Standard Deviation Which curve has the greater standard deviation?

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Example: Interpreting Graphs The scaled test scores for the New York State Grade 8 Mathematics Test are normally distributed. The normal curve shown below represents this distribution. What is the mean test score? Estimate the standard deviation.

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The Standard Normal Distribution Standard normal distribution A normal distribution with a mean of 0 and a standard deviation of 1.

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The Standard Normal Distribution If each data value of a normally distributed random variable x is transformed into a z-score, the result will be the standard normal distribution.

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Properties of the Standard Normal Distribution The cumulative area is close to 0 for z-scores close to z = –3.49. The cumulative area increases as the z-scores increase.

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Properties of the Standard Normal Distribution The cumulative area for z = 0 is 0.5000. The cumulative area is close to 1 for z-scores close to z = 3.49.

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Example: Using The Standard Normal Table Find the cumulative area that corresponds to a z-score of 1.15.

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Example: Using The Standard Normal Table Find the cumulative area that corresponds to a z-score of –0.24.

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Finding Areas Under the Standard Normal Curve Sketch the standard normal curve and shade the appropriate area under the curve. Find the area by following the directions for each case shown. To find the area to the left of z, find the area that corresponds to z in the Standard Normal Table.

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Finding Areas Under the Standard Normal Curve To find the area to the right of z, use the Standard Normal Table to find the area that corresponds to z. Then subtract the area from 1.

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Finding Areas Under the Standard Normal Curve To find the area between two z-scores, find the area corresponding to each z-score in the Standard Normal Table. Then subtract the smaller area from the larger area.