1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001....

41
1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman MARIO F. TRIOLA EIGHTH EDITION ELEMENTARY STATISTICS pter 5 Normal Probability Distribut

Transcript of 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001....

Page 1: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

1Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICSChapter 5 Normal Probability Distributions

Page 2: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

2Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Continuous random variable

Normal distribution

Overview5-1

Page 3: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

3Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Continuous random variable

Normal distributionCurve is bell shaped

and symmetric

µScore

Overview5-1

Figure 5-1

Page 4: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

4Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Continuous random variable

Normal distributionCurve is bell shaped

and symmetric

µScore

Formula 5-1

Overview5-1

Figure 5-1

x - µ 2

y =

12

e 2 p

( )

Page 5: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

5Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

5-2

The Standard Normal Distribution

Page 6: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

6Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Uniform Distribution a probability distribution in which the

continuous random variable values are spread evenly over the range of     

possibilities; the graph results in a     rectangular shape.

Definitions

Page 7: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

7Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Density Curve (or probability density   function)

  the graph of a continuous probability   distribution

Definitions

Page 8: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

8Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Density Curve (or probability density function) :

The graph of a continuous probability distribution

Definitions

1. The total area under the curve must equal 1.

2. Every point on the curve must have a vertical height that is 0 or greater.

Page 9: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

9Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Because the total area under the density curve is equal to 1,

there is a correspondence between area and probability.

Page 10: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

10Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Times in First or Last Half Hours

Figure 5-3

Page 11: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

11Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Heights of Adult Men and Women

Women:µ = 63.6 = 2.5 Men:

µ = 69.0 = 2.8

69.063.6Height (inches)

Figure 5-4

Page 12: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

12Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

DefinitionStandard Normal Deviation

a normal probability distribution that has a

mean of 0 and a standard deviation of 1

Page 13: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

13Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

DefinitionStandard Normal Deviation

a normal probability distribution that has a

mean of 0 and a standard deviation of 1

0 1 2 3-1-2-3 0 z = 1.58

Figure 5-5 Figure 5-6

Area = 0.3413 Area found in

Table A-2

0.4429

Score (z )

Page 14: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

14Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Table A-2 Standard Normal Distribution

µ = 0 = 1

0 x

z

Page 15: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

15Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

.0239

.0636

.1026

.1406

.1772

.2123

.2454

.2764

.3051

.3315

.3554

.3770

.3962

.4131

.4279

.4406

.4515

.4608

.4686

.4750

.4803

.4846

.4881

.4909

.4931

.4948

.4961

.4971

.4979

.4985

.4989

0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.02.12.22.32.42.52.62.72.82.93.0

.0000

.0398

.0793

.1179

.1554

.1915

.2257

.2580

.2881

.3159

.3413

.3643

.3849

.4032

.4192

.4332

.4452

.4554

.4641

.4713

.4772

.4821

.4861

.4893

.4918

.4938

.4953

.4965

.4974

.4981

.4987

.0040

.0438

.0832

.1217

.1591

.1950

.2291

.2611

.2910

.3186

.3438

.3665

.3869

.4049

.4207

.4345

.4463

.4564

.4649

.4719

.4778

.4826

.4864

.4896

.4920

.4940

.4955

.4966

.4975

.4982

.4987

.0080

.0478

.0871

.1255

.1628

.1985

.2324

.2642

.2939

.3212

.3461

.3686

.3888

.4066

.4222

.4357

.4474

.4573

.4656

.4726

.4783

.4830

.4868

.4898

.4922

.4941

.4956

.4967

.4976

.4982

.4987

.0120

.0517

.0910

.1293

.1664

.2019

.2357

.2673

.2967

.3238

.3485

.3708

.3907

.4082

.4236

.4370

.4484

.4582

.4664

.4732

.4788

.4834

.4871

.4901

.4925

.4943

.4957

.4968

.4977

.4983

.4988

.0160

.0557

.0948

.1331

.1700

.2054

.2389

.2704

.2995

.3264

.3508

.3729

.3925

.4099

.4251

.4382

.4495

.4591

.4671

.4738

.4793

.4838

.4875

.4904

.4927

.4945

.4959

.4969

.4977

.4984

.4988

.0199

.0596

.0987

.1368

.1736

.2088

.2422

.2734

.3023

.3289

.3531

.3749

.3944

.4115

.4265

.4394

.4505

.4599

.4678

.4744

.4798

.4842

.4878

.4906

.4929

.4946

.4960

.4970

.4978

.4984

.4989

.0279

.0675

.1064

.1443

.1808

.2157

.2486

.2794

.3078

.3340

.3577

.3790

.3980

.4147

.4292

.4418

.4525

.4616

.4693

.4756

.4808

.4850

.4884

.4911

.4932

.4949

.4962

.4972

.4979

.4985

.4989

.0319

.0714

.1103

.1480

.1844

.2190

.2517

.2823

.3106

.3365

.3599

.3810

.3997

.4162

.4306

.4429

.4535

.4625

.4699

.4761

.4812

.4854

.4887

.4913

.4934

.4951

.4963

.4973

.4980

.4986

.4990

.0359

.0753

.1141

.1517

.1879

.2224

.2549

.2852

.3133

.3389

.3621

.3830

.4015

.4177

.4319

.4441

.4545

.4633

.4706

.4767

.4817

.4857

.4890

.4916

.4936

.4952

.4964

.4974

.4981

.4986

.4990

*

*

.00 .01 .02 .03 .04 .05 .06 .07 .08 .09zTable A-2 Standard Normal (z) Distribution

Page 16: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

16Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

To find:

z Scorethe distance along horizontal scale of the standard normal distribution; refer to the leftmost column and top row of Table A-2

Area

the region under the curve; refer to the values in the body of Table A-2

Page 17: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

17Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees.

Page 18: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

18Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees.

0 1.58

P ( 0 < x < 1.58 ) =

Page 19: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

19Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.02.12.22.32.42.52.62.72.82.93.0

.0000

.0398

.0793

.1179

.1554

.1915

.2257

.2580

.2881

.3159

.3413

.3643

.3849

.4032

.4192

.4332

.4452

.4554

.4641

.4713

.4772

.4821

.4861

.4893

.4918

.4938

.4953

.4965

.4974

.4981

.4987

.0040

.0438

.0832

.1217

.1591

.1950

.2291

.2611

.2910

.3186

.3438

.3665

.3869

.4049

.4207

.4345

.4463

.4564

.4649

.4719

.4778

.4826

.4864

.4896

.4920

.4940

.4955

.4966

.4975

.4982

.4987

.0080

.0478

.0871

.1255

.1628

.1985

.2324

.2642

.2939

.3212

.3461

.3686

.3888

.4066

.4222

.4357

.4474

.4573

.4656

.4726

.4783

.4830

.4868

.4898

.4922

.4941

.4956

.4967

.4976

.4982

.4987

.0120

.0517

.0910

.1293

.1664

.2019

.2357

.2673

.2967

.3238

.3485

.3708

.3907

.4082

.4236

.4370

.4484

.4582

.4664

.4732

.4788

.4834

.4871

.4901

.4925

.4943

.4957

.4968

.4977

.4983

.4988

.0160

.0557

.0948

.1331

.1700

.2054

.2389

.2704

.2995

.3264

.3508

.3729

.3925

.4099

.4251

.4382

.4495

.4591

.4671

.4738

.4793

.4838

.4875

.4904

.4927

.4945

.4959

.4969

.4977

.4984

.4988

.0199

.0596

.0987

.1368

.1736

.2088

.2422

.2734

.3023

.3289

.3531

.3749

.3944

.4115

.4265

.4394

.4505

.4599

.4678

.4744

.4798

.4842

.4878

.4906

.4929

.4946

.4960

.4970

.4978

.4984

.4989

.0239

.0636

.1026

.1406

.1772

.2123

.2454

.2764

.3051

.3315

.3554

.3770

.3962

.4131

.4279

.4406

.4515

.4608

.4686

.4750

.4803

.4846

.4881

.4909

.4931

.4948

.4961

.4971

.4979

.4985

.4989

.0279

.0675

.1064

.1443

.1808

.2157

.2486

.2794

.3078

.3340

.3577

.3790

.3980

.4147

.4292

.4418

.4525

.4616

.4693

.4756

.4808

.4850

.4884

.4911

.4932

.4949

.4962

.4972

.4979

.4985

.4989

.0319

.0714

.1103

.1480

.1844

.2190

.2517

.2823

.3106

.3365

.3599

.3810

.3997

.4162

.4306

.4429

.4535

.4625

.4699

.4761

.4812

.4854

.4887

.4913

.4934

.4951

.4963

.4973

.4980

.4986

.4990

.0359

.0753

.1141

.1517

.1879

.2224

.2549

.2852

.3133

.3389

.3621

.3830

.4015

.4177

.4319

.4441

.4545

.4633

.4706

.4767

.4817

.4857

.4890

.4916

.4936

.4952

.4964

.4974

.4981

.4986

.4990

*

*

.00 .01 .02 .03 .04 .05 .06 .07 .08 .09zTable A-2 Standard Normal (z) Distribution

Page 20: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

20Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees.

0 1.58

Area = 0.4429

P ( 0 < x < 1.58 ) = 0.4429

Page 21: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

21Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees.

The probability that the chosen thermometer will measure freezing water between 0 and 1.58 degrees is 0.4429.

0 1.58

Area = 0.4429

P ( 0 < x < 1.58 ) = 0.4429

Page 22: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

22Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees.

There is 44.29% of the thermometers with readings between 0 and 1.58 degrees.

0 1.58

Area = 0.4429

P ( 0 < x < 1.58 ) = 0.4429

Page 23: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

23Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Using Symmetry to Find the Area to the Left of the Mean

NOTE: Although a z score can be negative, the area under the curve (or the corresponding probability)

can never be negative.

(a) (b)

Because of symmetry, these areas are equal.

Equal distance away from 0

0.4925 0.4925

0 0

z = 2.43z = - 2.43

Figure 5-7

Page 24: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

24Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water, and if one thermometer is randomly selected, find the probability that it reads freezing water between -2.43 degrees and 0 degrees.

The probability that the chosen thermometer will measure freezing water between -2.43 and 0 degrees is 0.4925.

-2.43 0

Area = 0.4925P ( -2.43 < x < 0 ) = 0.4925

Page 25: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

25Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

The Empirical RuleStandard Normal Distribution: µ = 0 and = 1

Page 26: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

26Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x - s x x + s

68% within1 standard deviation

34% 34%

The Empirical RuleStandard Normal Distribution: µ = 0 and = 1

Page 27: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

27Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x - 2s x - s x x + 2sx + s

68% within1 standard deviation

34% 34%

95% within 2 standard deviations

13.5% 13.5%

The Empirical RuleStandard Normal Distribution: µ = 0 and = 1

Page 28: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

28Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x - 3s x - 2s x - s x x + 2s x + 3sx + s

68% within1 standard deviation

34% 34%

95% within 2 standard deviations

99.7% of data are within 3 standard deviations of the mean

0.1% 0.1%

2.4% 2.4%

13.5% 13.5%

The Empirical RuleStandard Normal Distribution: µ = 0 and = 1

Page 29: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

29Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Probability of Half of a Distribution

0

0.5

Page 30: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

30Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Area to the Right of z = 1.27

0

0.3980

Value foundin Table A-2

This area is 0.5 - 0.3980 = 0.1020

z = 1.27

Figure 5-8

Page 31: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

31Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding the Area Between z = 1.20 and z = 2.30

0

0.3849

0.4893 (from Table A-2 with z = 2.30)

Area A is 0.4893 - 0.3849 =

0.1044

z = 1.20

Az = 2.30

Figure 5-9

Page 32: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

32Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

P(a < z < b) denotes the probability that the z score is

between a and b

P(z > a) denotes the probability that the z score is

greater than a

P (z < a) denotes the probability that the z score is

less than a

Notation

Page 33: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

33Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Figure 5-10 Interpreting Area CorrectlyAdd to

0.5

0.5

x

‘greater than x’

‘at least x’

‘more than x’

‘not less than x’

x

Subtractfrom 0.5

Page 34: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

34Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Figure 5-10 Interpreting Area CorrectlyAdd to

0.5

0.5

xAdd to

0.5

0.5

x

‘greater than x’

‘at least x’

‘more than x’

‘not less than x’

x

Subtractfrom 0.5

x

Subtractfrom 0.5

‘less than x’

‘at most x’

‘no more than x’

‘not greater than x’

Page 35: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

35Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Figure 5-10 Interpreting Area CorrectlyAdd to

0.5

0.5

xAdd to

0.5

0.5

x

‘greater than x’

‘at least x’

‘more than x’

‘not less than x’

x

Subtractfrom 0.5

x

Subtractfrom 0.5

x1 x2

Add

‘less than x’

‘at most x’

‘no more than x’

‘not greater than x’

‘between x1 and x2’A B

UseA = C - B

x1 x2

C

Page 36: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

36Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Finding a z - score when given a probabilityUsing Table A-2

1. Draw a bell-shaped curve, draw the centerline, and identify the region under the curve that corresponds to the given probability. If that region is not bounded by the centerline, work with a known region that is bounded by the centerline.

2. Using the probability representing the area bounded by the centerline, locate the closest probability in the body of Table A-2 and identify the corresponding z score.

3. If the z score is positioned to the left of the centerline, make it a negative.

Page 37: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

37Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

0

0.45

z

0.50

95% 5%

5% or 0.05

Finding z Scores when Given Probabilities

FIGURE 5-11 Finding the 95th Percentile( z score will be positive )

Page 38: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

38Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

0

0.45

1.645

0.50

95% 5%

5% or 0.05

Finding z Scores when Given Probabilities

FIGURE 5-11 Finding the 95th Percentile(z score will be positive)

Page 39: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

39Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

0

0.40

z

0.10

90%

FIGURE 5-12 Finding the 10th Percentile

Bottom 10%

10%

(z score will be negative)

Finding z Scores when Given Probabilities

Page 40: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

40Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

0

0.40-1.28

0.10

90%

FIGURE 5-12 Finding the 10th Percentile

Bottom 10%

10%

Finding z Scores when Given Probabilities

(z score will be negative)

Page 41: 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION.

41Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Assignment

• Page 240: 1-36 all