How do I use normal distributions in finding probabilities?

• How do I use normal distributions in finding probabilities?

7.4 Use Normal Distributions

Standard Deviation of a Data Set

x

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2

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2

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3

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3

68%

A normal distribution with mean x and standard deviation has these properties:• The total area under the related normal curve is ____.1• About ___% of the area lies within 1 standard

deviation of the mean.68

• About ___% of the area lies within 2 standard deviation of the mean.

95

• About _____% of the area lies within 3 standard deviation of the mean.

7.99

95%

99.7%

34% 34%

13.5% 13.5%2.35% 2.35%

0.15%

x

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2

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3

0.15%

7.4 Use Normal DistributionsExample 1 Find a normal probabilityFind a normal probability

A normal distribution has a mean A normal distribution has a mean xx and standard deviation and standard deviation . For a . For a randomly selected randomly selected xx-value from -value from the distribution, findthe distribution, find x

x –

x +

x –

2

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2

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3

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3

2 xxxPSolutionThe probability that a randomly selected x-value lies between _______ and _________ is the shaded area under the normal curve. Therefore:

________________2 xxxP 34.0 34.0 135.0_______ 815.0

x 2x

1. A normal distribution has mean x and

standard deviation . For a randomly

selected x-value from the distribution,

find

7.4 Use Normal DistributionsCheckpoint. Complete the following exercise.Checkpoint. Complete the following exercise.

.xxP

x

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x +

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2

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3

50.0 34.0 16.0

%____%____%___

7.4 Use Normal DistributionsExample 2 Interpret normally distributed dataInterpret normally distributed data

The math scores of an exam for the The math scores of an exam for the state of Georgia are normally state of Georgia are normally distributed with a mean of 496 and a distributed with a mean of 496 and a standard deviation of 109. About what standard deviation of 109. About what percent of the test-takers received percent of the test-takers received scores between 387 and 605?scores between 387 and 605?SolutionThe scores of 387 and 605 represent ____ standard deviation on either side of the mean. So the percent of test-takers with scores between 387 and 605 is

34 34 68

one

496 605 714 823387278169

2. In Example 2, what percent of

the test-takers received scores

between 496 and 714?


496 605 714 823387278169

34%

13.5%

34 5.13 %5.47

__________________________

xx

z

7.4 Use Normal DistributionsExample 3 Use a Use a zz-score and the standard normal table-score and the standard normal table

In Example 2, find the probability that a randomly In Example 2, find the probability that a randomly selected test-taker received a math score of at most 630?selected test-taker received a math score of at most 630?SolutionSep 1 Find the z-score corresponding to an x-value of 630.

496630 109

2.1

___630 zPxP

7.4 Use Normal DistributionsExample 3 Use a Use a zz-score and the standard normal table-score and the standard normal table

In Example 2, find the probability that a randomly In Example 2, find the probability that a randomly selected test-taker received a math score of at most 630?selected test-taker received a math score of at most 630?SolutionSep 2 Use the standard normal table to find

2.1z .0 .1 .2

.0013 .0010 .0007

The table shows that P(z < ____) = _______.2.1 8849.So, the probability that a randomly selected test-taker received a math score of at most 630 is about ________.8849.

3. In Example 3, find the probability that a randomly selected test-taker received a math score of at most 620?


xx

z496620

1091.1

620xP 1.1 zP 8643.0

7.4 Use Normal Distributions

Pg. 277, 7.4 #1-21

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How do I use normal distributions in finding probabilities?