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**Section 9.3 The Normal Distribution**

Chapter 9 Statistics Section 9.3 The Normal Distribution

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**Continuous Distributions**

Recall that a histogram and its corresponding frequency polygon can be constructed from information obtained from a frequency distribution or a probability distribution.

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**Continuous Distributions**

A bank manager collects data to determine the amount of time to the nearest minute tellers spend on each transaction.

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**Continuous Distributions**

A distribution in which the outcomes can take any real number value within some interval is a continuous distribution. The graph of a continuous distribution is a curve. Distributions whose peak is not at the center are called skewed.

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Normal Distributions Many natural and social phenomena produce continuous probability distributions whose graphs can be approximated by bell-shaped curves. These kinds of distributions are called normal distributions and their graphs are called normal curves. For a normal distribution, the Greek letter µ (mu) is used to denote the mean, and σ (sigma) is used to denote the standard deviation.

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**Examples of Normal Distributions**

The smaller the standard deviation, the taller and narrower the curve will be. The larger the standard deviation, the wider and more flat the curve will be.

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**Properties of Normal Distributions**

The peak occurs directly above the mean µ. The curve is symmetric about the vertical line through the mean. The curve never touches the x-axis – it extends indefinitely in both directions. The area under the curve (and above the horizontal axis) is always 1. (Sum of the probabilities in a probability distribution is always 1.)

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**Determining Probabilities of a Normal Distribution**

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**Determining Probabilities of a Normal Distribution**

To use normal curves effectively, we must be able to calculate areas under portions of these curves. These calculations have already been done for the normal curve with mean µ = 0 and standard deviation σ = 1. (This is the standard normal curve. The table of these calculations is found in the Appendix of your textbook.

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**Area Under a Normal Curve Table**

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Standard Normal Curve The horizontal axis of the standard normal curve is usually labeled z. When calculating normal probability, always draw a normal curve with the mean and z-scores every time.

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Example 1 Find the percent of the area under a normal curve between the mean and the given number of standard deviations from the mean. a.) b.)

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Example 2 Find the percent of the total area under the standard normal curve between each pair of z-scores. a.) and b.) and 1.17

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**Example 3 Find a z-score satisfying the following conditions.**

a.) 45% of the total area is to the left of z. b.) 20% of the total area is to the right of z.

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Important!! The key to finding areas under any normal curve is to express each number x on the horizontal axis in terms of a standard deviation above or below the mean. The z-score for x is the number of standard deviations that x lies from the mean (positive if x is above the mean, negative if x is below the mean).

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**Converting a Data Value X to a Z-score**

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**Importance of Z-scores**

By converting data values to z-scores and using the table for the standard normal curve, we can find areas under any normal curve. Since the areas are probabilities, we can now how handle a variety of a applications. FUN!!

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**Applications of the Standard Normal Curve**

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Example 4 A certain type of light bulb has an average life of 500 hours, with a standard deviation of 100 hours. The length of life of the bulb can be closely approximated by a normal curve. An amusement park buys and installs 10,000 such bulbs. Find the total number that can be expected to last for the following periods of time a.) at least 500 hours b.) between 650 and 780 hours

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Example 5 A machine that fills quart orange juice cartons is set to fill them with 32.1 oz. If the actual contents of the cartons vary normally, with a standard deviation of 0.1 oz, what percent of the cartons contains less than a quart (32 oz)?

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Example 6 On standard IQ tests, the mean is 100, with a standard deviation of 15. The results are very close to fitting a normal curve. Suppose an IQ test is given to a very large group of people. Find the percent of those people whose IQ scores are more than 130.

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