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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
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
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
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
( )
5Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5-2
The Standard Normal Distribution
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
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
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.
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.
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
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
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
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 )
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
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
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
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.
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 ) =
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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’
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
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.
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 )
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)
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
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)
41Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Assignment
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