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Section 5.1 Introduction to Normal Distributions and the Standard Normal Distribution

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**Properties of a Normal Distribution**

Guidelines: Properties of a Normal Distribution: A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve. A normal distribution has the following properties. The mean, median, and mode are equal. The normal curve is bell shaped and is symmetric about the mean. The total are under the normal curve is equal to one. The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean. 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 inflection points.

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**Example 1 Which normal curve has a greater mean? **

Which normal curve has a greater standard deviation?

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Example 2 The heights (in feet) of fully grown white oak trees are normally distributed. The normal curve shown below represents this distribution. What is the mean height of a fully grown oak tree? Estimate the standard deviation of this normal distribution.

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**The Standard Normal Distribution**

Definition 1: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

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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. The cumulative area for z = 0 is The cumulative area is close to 1 for z-scores close to z = 3.49

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**Standard Normal Distribution**

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Example 3 Find the cumulative area that corresponds to a z-score of 1.15.

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TOTD Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences.

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Example 3 Find the cumulative area that corresponds to a z-score of

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Example 3 Find the cumulative area that corresponds to a z-score of

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**Guidelines: 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 corresponding to z in the Standard Normal Table. 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. 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.

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Example 4 Find the area under the standard normal curve to the left of z =

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Example 4 Find the area under the standard normal curve to the right of z = 1.06.

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Example 4 Find the area under the standard normal curve between z = -1.5 and z = 1.25.

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Example 4 Find the area under the standard normal curve between z = and z =

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**TOTD Find the indicated area under the standard normal curve.**

To the right of z = 1.645 Between z = and z = 0

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