chapter 6 the normal distribution (the original slide set from the bluman text has been trimmed and...

Chapter 6

The Normal Distribution(The original slide set from the Bluman text has been trimmed and augmented with extra explanations and TI-84 examples. The Bluman slides have the footer and rose background.)

© McGraw-Hill, Bluman, 5th ed., Chapter 6 1

6.1 Normal Distributions Many continuous variables have distributions

that are bell-shaped and are called approximately normally distributed variables.

The theoretical curve, called the bell curve or the Gaussian distribution, can be used to study many variables that are not normally distributed but are approximately normal.

Bluman, Chapter 6 2

Normal Distributions

2 2( ) (2 )

2

Xe

y

Bluman, Chapter 6 3

The mathematical equation for the a normal distribution is:

2.718

3.14

where

e

population mean

population standard deviation

Normal Distributions The shape and position of the normal

distribution curve depend on two parameters, the mean and the standard deviation.

Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable’s mean and standard deviation.

Bluman, Chapter 6 4

Normal Distributions - different onesfor different values of and

Bluman, Chapter 6 5

Normal Distribution Properties The normal distribution curve is bell-shaped. The mean, median, and mode are equal and

located at the center of the distribution. The normal distribution curve is unimodal (i.e.,

it has only one mode). The curve is symmetrical about the mean,

which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.

Bluman, Chapter 6 6

Normal Distribution Properties The curve is continuous—i.e., there are no

gaps or holes. For each value of X, here is a corresponding value of Y.

The curve never touches the x axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis—but it gets increasingly closer.

Bluman, Chapter 6 7

Normal Distribution Properties They said “it never meets the x axis—but it gets

increasingly closer.” Example: for the standard normal distribution

where , when And when

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Normal Distribution Properties The total area under the normal distribution

curve is equal to 1.00 or 100%. The area under the normal curve that lies within

one standard deviation of the mean is approximately 0.68 (68%).

two standard deviations of the mean is approximately 0.95 (95%).

three standard deviations of the mean is approximately 0.997 ( 99.7%).

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Normal Distribution Properties

Bluman, Chapter 6 10

“The Empirical Rule”

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Standard Normal Distribution Since each normally distributed variable has its

own mean and standard deviation, the shape and location of these curves will vary. In practical applications, one would have to have a table of areas under the curve for each variable. To simplify this, statisticians use the standard normal distribution.

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

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The Standard Normal Distribution is our favorite This one is the most special of all of them

The mean: The standard deviation:

Horizontal Total area between curve and axis = 1.000

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z value (Standard Value)The z value is the number of standard deviations that a particular X value is away from the mean. The formula for finding the z value is:

Bluman, Chapter 6 14

value - mean

standard deviationz

Xz

x valueGoing the other way:

If you know the z value

and you need to find the x value,

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Area ProblemsKEY CONNECTION: PROBABILITY is AREA AREA is PROBABILITY

HOW THIS AFFECTS YOUR LIFE: You want to answer Probability questions You find the answers by finding Areas of

regions underneath the Normal Curve.

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Three kinds of area problems

“What is the area to the left of some ?” “What is the area to the right of some ?” “What is the area between and ?”

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How do you find these areas?

1. By using a printed table that gives areas to the left of and

2. Or by using the TI-84 normalcdf(z1,z2) function.

3. Or by using Excel’s =NORM.S.DIST(z,TRUE) function

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Area under the Standard Normal Distribution Curve1. To the left of any z value:

Look up the z value in the table and use the area given.

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Area under the Standard Normal Distribution Curve2. To the right of any z value:

Look up the z value and subtract the area from 1.

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Area under the Standard Normal Distribution Curve3. Between two z values:

Look up both z values and subtract the corresponding areas.

Bluman, Chapter 6 21

TI-84 Methods

To get normalcdf() 2ND DISTR

It’s on the VARS key

2:normalcdf(

Beware! You do NOT want to use 1:normalpdf( in most cases in this course.

Area between two z vals. Normalcdf(zLow, zHigh)

Example: Area between and

Agrees with the Empirical Rule (68/95/99.7) !

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TI-84 Methods

Area to the left of z normalcdf(-99,z) Example: Area to the left

of z=1.23

Area to the right of z normalcdf(z,99) Example: Area to the right

of z=1.23

Observe: area to left + area to right = 1.0000000

Not a coincidence!!!!

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Example 6-1: Area under the Curve

Find the area to the left of z = 1.99.

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The value in the 1.9 row and the .09 column of Table E is .9767. The area is .9767.

Example 6-1: Area under the Curve

Find the area to the left of z = 1.99.

They got 0.9767 using the printed table.

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Example 6-2: Area under the Curve

Find the area to right of z = -1.16.

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The value in the -1.1 row and the .06 column of Table E is .1230. The area is 1 - .1230 = .8770.

Example 6-2: Area under the Curve

Find the area to right of z = -1.16.

They used the printed table, which only gives areas to the left, so they had to subtract,

1 -0.1230 = 0.8770.

The TI-84 normalcdf() was more direct.

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Example 6-3: Area under the Curve

Find the area between z = 1.68 and z = -1.37.

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The values for z = 1.68 is .9535 and for z = -1.37 is .0853. The area is .9535 - .0853 = .8682.

Example 6-3: Area under the Curve

Find the area between z = 1.68 and z = -1.37.

With the printed tables, they had to do two lookups and subtract the results to get .8682

The TI-84 normalcdf() was more direct.

Bluman, Chapter 6 29

Problems that work backwards

They give you the area You have to work backwards to find the z

score.

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Example of a backwards problem What z value divides the area under the

standard normal curve so that the area to the left of that z is 0.7123? And what is the area to the right of that z score?

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How to solve it using the printed table

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The z value is 0.56.

The area to the left is .7123. Then look for that value inside Table E.

Backwards problem using TI-84

invNorm( is the tool 2ND DISTR again 3:invNorm( Stands for “Inverse

Normal” Tell him area to the left He responds with the z

score.

Example: area 0.7123

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If they give you area to the right

Example: Find z so area to the right of z is 0.7500

But Tables & TI-84 deal with areas to the left.

COMPLEMENT: If area to the right is 0.7500, are to the left is 1 – 0.7500

So we seek the z that has 0.2500 to its left.

How to solve it

In table: Lookup 0.2500 in table. If it’s not there, take the

closest value. Read out to find z again. With TI-84 invNorm(

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Backwards Area-Between

“Find the z scores that delimit the middle 80% of the area” DRAW A PICTURE!!!

0.8000 is in the middle So 1.0000 – 0.8000 = 0.2000 in two tails 0.2000 ÷ 2 = 0.1000 in each tail

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Backwards Area-Between

Using Printed Table Look deep in table for

closest match to 0.1000 Read out to find the

negative z on the left. Because of symmetry, the

positive z on the right is the opposite of that value.

Using TI-84 invNorm(.1000) Because of symmetry, the

positive z on the right is the opposite of that value.

Answers: and

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Confirming this with TI-84

Using normalcdf() normalcdf(-1.28,1.28)

should give about area .8000 they asked for

More decimals for more precision

Using DRAW 2ND DRAW 1:ClrDraw 2ND DISTR Right arrow to DRAW 1:ShadeNorm(z1,z2)

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Confirming this with TI-84

WINDOW settings It probably won’t turn out

well on its own. You may need to do some thinking.

What we did for this one:

Using DRAW 2ND DRAW 1:ClrDraw 2ND DISTR Right arrow to DRAW 1:ShadeNorm(z1,z2)

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