7.6 The Normal Distribution

1. Normal Curve2. Normally Distributed Outcomes3. Properties of Normal Curve4. Standard Normal Curve5. The Normal Distribution 6. Percentile7. Probability for General Normal

Distribution

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Normal Curve The bell-shaped curve, as shown below, is

call a normal curve.

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Normally Distributed Outcomes

Examples of experiments that have normally distributed outcomes:

1. Choose an individual at random and observe his/her IQ.

2. Choose a 1-day-old infant and observe his/her weight.

3. Choose a leaf at random from a particular tree and observe its length.

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Properties of Normal Curve

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Example Properties of Normal Curve

A certain experiment has normally distributed outcomes with mean equal to 1. Shade the region corresponding to the probability that the outcome

(a) lies between 1 and 3; (b) lies between 0 and 2; (c) is less than .5; (d) is greater than 2.

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Example Properties of Normal Curve (2)

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Standard Normal Curve

The equation of the normal curve is

21

21

2where 3.1416 and 2.7183.

x

y e

e

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The standard normal curve has 0 and 1.

The Normal Distribution

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A(z) is the area under the standard normal curve to the left of a normally distributed random variable z.

Example The Normal Distribution

Use the normal distribution table to determine the area corresponding to

(a) z < -.5; (b) 1< z < 2; (c) z > 1.5.

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Example The Normal Distribution (2)

(a) A(-.5) = .3085

(b) A(2) - A(1) = .9772 - 8413

= .1359

(c) 1 - A(1.5) = 1 - .9332 = .0668

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Percentile

If a score S is the pth percentile of a normal distribution, then p% of all scores fall below S, and (100 - p)% of all scores fall above S. The pth percentile is written as zp.

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Example Percentile

What is the 95th percentile of the standard normal distribution?

In the normal distribution, find the value of z such that A(z) = .95.

A(1.65) = .9506 Therefore, z95 = 1.65.

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Probability for General Normal Distribution

If X is a random variable having a normal distribution with mean and standard deviation then

where Z has the standard normal distribution and A(z) is the area under that distribution to the left of z.

Pr( ) Pr

and Pr( ) Pr

a b b aa X b Z A A

x xX x Z A

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,

Example Probability Normal Distribution

Find the 95th percentile of infant birth weights if infant birth weights are normally distributed with = 7.75 and = 1.25 pounds.

The value for the standard normal random variable is z95 = 1.65.

Then x95 = 7.75 + (1.65)(1.25) = 9.81 pounds.

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Summary Section 7.6 - Part 1

A normal curve is identified by its mean ( ) and its standard deviation ( ). The standard normal curve has = 0 and = 1. Areas of the region under the standard normal curve can be obtained with the aid of a table or graphing calculator.

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Summary Section 7.6 - Part 2

A random variable is said to be normally distributed if the probability that an outcome lies between a and b is the area of the region under a normal curve from x = a to x = b. After the numbers a and b are converted to standard deviations from the mean, the sought-after probability can be obtained as an area under the standard normal curve.

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