Chapter 9Chapter 9StatisticsStatistics

Section 9.3Section 9.3

The Normal DistributionThe Normal Distribution

Continuous DistributionsContinuous Distributions Recall that a histogram and its Recall that a histogram and its

corresponding frequency polygon can corresponding frequency polygon can be constructed from information be constructed from information obtained from a frequency distribution obtained from a frequency distribution or a probability distribution.or a probability distribution.

Continuous DistributionsContinuous Distributions A bank manager collects data to determine the A bank manager collects data to determine the

amount of time to the nearest minute tellers amount of time to the nearest minute tellers spend on each transaction.spend on each transaction.

Continuous DistributionsContinuous Distributions A distribution in which the outcomes A distribution in which the outcomes

can take any real number value can take any real number value within some interval is a within some interval is a continuous continuous distributiondistribution. .

The graph of a continuous The graph of a continuous distribution is a curve.distribution is a curve.

Distributions whose peak is not at Distributions whose peak is not at the center are called the center are called skewedskewed..

Normal DistributionsNormal Distributions Many natural and social phenomena Many natural and social phenomena

produce continuous probability produce continuous probability distributions whose graphs can be distributions whose graphs can be approximated by bell-shaped curves.approximated by bell-shaped curves.

These kinds of distributions are called These kinds of distributions are called normal distributionsnormal distributions and their graphs are and their graphs are called called normal curvesnormal curves. .

For a normal distribution, the Greek letter For a normal distribution, the Greek letter µµ (mu) is used to denote the mean, and (mu) is used to denote the mean, and σσ (sigma) is used to denote the standard (sigma) is used to denote the standard deviation.deviation.

Examples of Normal Examples of Normal DistributionsDistributions

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.

Properties of Normal Properties of Normal DistributionsDistributions

The peak occurs directly above the mean The peak occurs directly above the mean µ.µ.

The curve is symmetric about the vertical line The curve is symmetric about the vertical line through the mean.through the mean.

The curve never touches the The curve never touches the xx-axis – it extends -axis – it extends indefinitely in both directions.indefinitely in both directions.

The area under the curve (and above the The area under the curve (and above the horizontal axis) is always 1. (Sum of the horizontal axis) is always 1. (Sum of the probabilities in a probability distribution is always probabilities in a probability distribution is always 1.)1.)

Determining Probabilities of a Determining Probabilities of a Normal DistributionNormal Distribution

Determining Probabilities of a Determining Probabilities of a Normal DistributionNormal Distribution

To use normal curves effectively, we To use normal curves effectively, we must be able to calculate areas under must be able to calculate areas under portions of these curves.portions of these curves.

These calculations have already been These calculations have already been done for the normal curve with mean done for the normal curve with mean µµ = = 0 and standard deviation 0 and standard deviation σσ = 1. (This is = 1. (This is the the standard normal curvestandard normal curve..

The table of these calculations is found in The table of these calculations is found in the Appendix of your textbook.the Appendix of your textbook.

Area Under a Normal Curve Area Under a Normal Curve TableTable

Standard Normal CurveStandard Normal Curve

The horizontal axis of the standard The horizontal axis of the standard normal curve is usually labeled normal curve is usually labeled zz..

When calculating normal probability, When calculating normal probability, always draw a normal curve with the always draw a normal curve with the mean and mean and zz-scores every time.-scores every time.

Example 1Example 1 Find the percent of the area under a Find the percent of the area under a

normal curve between the mean and normal curve between the mean and the given number of standard the given number of standard deviations from the mean.deviations from the mean.

a.) 1.87a.) 1.87 b.) -0.95b.) -0.95

Example 2Example 2 Find the percent of the total area under Find the percent of the total area under

the standard normal curve between each the standard normal curve between each pair of pair of zz-scores.-scores.

a.) 1.05 and 2.46a.) 1.05 and 2.46 b.) -2.15 and b.) -2.15 and 1.171.17

Example 3Example 3

Find a Find a zz-score satisfying the following -score satisfying the following conditions.conditions.

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

b.) 20% of the total area is to the right of b.) 20% of the total area is to the right of zz..

Important!!Important!!

The key to finding areas under The key to finding areas under anyany normal curve is to express each normal curve is to express each number number xx on the horizontal axis in on the horizontal axis in terms of a standard deviation above or terms of a standard deviation above or below the mean.below the mean.

The The zz-score-score for for xx is the number of is the number of standard deviations that standard deviations that x x lies from the lies from the mean (positive if mean (positive if xx is above the mean, is above the mean, negative if negative if xx is below the mean). is below the mean).

Converting a Data Value Converting a Data Value XX to a to a ZZ-score-score

Importance of Importance of ZZ-scores-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!!

Applications of the Standard Applications of the Standard Normal CurveNormal Curve

Example 4Example 4 A certain type of light bulb has an A certain type of light bulb has an

average life of 500 hours, with a average life of 500 hours, with a standard deviation of 100 hours. The standard deviation of 100 hours. The length of life of the bulb can be closely length of life of the bulb can be closely approximated by a normal curve. An approximated by a normal curve. An amusement park buys and installs amusement park buys and installs 10,000 such bulbs. Find the total 10,000 such bulbs. Find the total number that can be expected to last for number that can be expected to last for the following periods of timethe following periods of time

a.) at least 500 hoursa.) at least 500 hours

b.) between 650 and 780 hoursb.) between 650 and 780 hours

Example 5Example 5 A machine that fills quart orange juice A machine that fills quart orange juice

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

Example 6Example 6 On standard IQ tests, the mean is 100, On standard IQ tests, the mean is 100,

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