Download lect5-1 - Rutgers Statistics

Normal Probability
Distributions
Chapter 5
M A R I O F. T R I O L A
Copyright © 1998, Triola, Elementary Statistics
Copyright © 1998, Triola, Elementary Statistics
Addison
Wesley
Longman
Addison
Wesley Longman
1
Chapter 5
Normal Probability Distributions
5-1 Overview
5-2 The Standard Normal Distribution
5-3 & 5-4 Nonstandard Normal Distributions
(Finding Probabilities & Finding Scores)
5-5 The Central Limit Theorem
5-6 Normal Distributions as Approximation
to Binomial Distribution
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2
5-2
Distributions of continuous
random variables
The Uniform Distribution
&
The Standard Normal
Distribution
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3
Uniform Distribution
Definition
a probability distribution in which every
value of the random variable is equally
likely
– The probability distribution is call a uniform
distribution
– The random variable is called a random
variable with a uniform distribution
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Definition
 Density Curve
a graph of a continuous probability distribution that
satisfies the following two properties:
1. the total area under the curve must be 1
2. every point on the curve must have a vertical
height that is 0 or greater
 A density curve is the graph of a continuous random
variable, so the area under the curve is 1 and there is a
correspondence between area and probability.
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5
Uniform Distributions
Shaded Area = L • W
= 3 • 0.2 = 0.6
P( 1º < x < 4º ) = 0.6
P(x )
P( x > 1º ) = 0.8
P(x )
3
0.2
0.2
x
0
0
1
2
3
4
5
Temperature (degrees Celsius)
x
0
0
1
2
3
4
5
Temperature (degrees Celsius)
Figure 5-3
Figure 5-2
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6
Different example:
Heights of Adult Men and Women
Women:
µ = 63.6
s = 2.5
Men:
µ = 69.0
s = 2.8
63.6
69.0
Height (inches)
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7
Normal
Distribution
Continuous random variable
 Normal distribution
Curve is bell shaped
and symmetric
µ
Score
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Normal
Distribution
Continuous random variable
 Normal distribution
Curve is bell shaped
and symmetric
µ
Score
Formula 5-1
y=
e
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1
2
s
(
x–µ
2
s
)
2
p
9
Definition
Standard Normal Distribution
a normal probability distribution that has a
mean of 0 and a standard deviation of 1
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Definition
Standard Normal Distribution
a normal probability distribution that has a
mean of 0 and a standard deviation of 1
•Area found in
•Table A-2
•Area = 0.3413
•0.4429
-3
-2
-1
0
1
Score (z )
2
3
Figure 5-5
0
z = 1.58
Figure 5-6
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Standard Normal Distribution
s=1
µ=0
?
0
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x
12
Table A-2
 Back left cover of text book
 Formula card
 Appendix
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13
Table A-2
Standard Normal Distribution
s=1
µ=0
0
x
z
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14
Table A-2 Standard Normal (z) Distribution
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
.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
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15
To find:
z Score
the distance along horizontal scale on
graph; refer to the leftmost column
(and top row) in Table A-2
Area (or probability)
the region under the curve; refer to the
numbers in the body of Table A-2
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16
Table A-2
Standard Normal Distribution
s=1
µ=0
.4495
0
Copyright © 1998, Triola, Elementary Statistics
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z = 1.64
17
Table A-2 Standard Normal (z) Distribution
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
.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
*
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Table A-2
Standard Normal Distribution
s=1
µ=0
.4949
0
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
z = 2.57
19
Table A-2 Standard Normal (z) Distribution
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
.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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
*
20
Using Symmetry to Find the Area
to the Left of the Mean
Because of symmetry, these areas are equal.
(a)
(b)
0.4925
0.4925
0
z = –2.43
0
Equal distance away from 0
z = 2.43
• NOTE: Although a z score can be negative, the area
under the curve (or the corresponding probabilities) can
never be negative.
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21
Table A-2 Standard Normal (z) Distribution
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
.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
*
*
*
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Addison Wesley Longman
22
The Empirical Rule
Standard Normal Distribution: µ = 0 and s = 1
0
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Addison Wesley Longman
23
The Empirical Rule
Standard Normal Distribution: µ = 0 and s = 1
68% within
1 standard deviation
0.340
–1
0.340
0
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Addison Wesley Longman
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24
The Empirical Rule
Standard Normal Distribution: µ = 0 and s = 1
95% within
2 standard deviations
68% within
1 standard deviation
0.340
0.340
0.135
–2
0.135
–1
0
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
1
2
25
The Empirical Rule
Standard Normal Distribution: µ = 0 and s = 1
99.7% of data are within 3 standard deviations of the mean
95% within
2 standard deviations
68% within
1 standard deviation
0.340
0.340
0.024
0.024
0.001
0.001
0.135
–3
–2
0.135
–1
0
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Addison Wesley Longman
1
2
3
26
Probability of Half of a Distribution
0.5
0
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Addison Wesley Longman
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Finding the Area to the Right of z = 1.27
Value found
in Table A-2
0.3980
0
This area is
0.5 – 0.3980 = 0.1020
z = 1.27
Figure 5-8
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
28
Finding the Area Between
z = 1.20 and z = 2.30
0.4893 (from Table A-2 with z = 2 .30)
0.3849
0
Area A is 0 .4893 – 0 .3849 = 0 .1044
A
z = 1.20 z = 2.30
Figure 5-9
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
29
Notation
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
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Addison Wesley Longman
30
Figure 5-10
Interpreting Area Correctly
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Figure 5-10
"greater than x"
"at least x"
"more than x"
"not less than x"
Interpreting Area Correctly
Add to
0. 5
Subtract
from
0. 5
0. 5
x
x
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Addison Wesley Longman
32
Figure 5-10
"greater than x"
"at least x"
"more than x"
"not less than x"
Interpreting Area Correctly
Add to
0. 5
Subtract
from
0. 5
0. 5
x
"less than x"
"at most x"
"no more than x"
"not greater than x"
x
Subtract
from
0.5
Add to
0. 5
0.5
x
x
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Addison Wesley Longman
33
Figure 5-10
"greater than x"
"at least x"
"more than x"
"not less than x"
Interpreting Area Correctly
Add to
0. 5
Subtract
from
0. 5
0. 5
x
"less than x"
"at most x"
"no more than x"
"not greater than x"
x
Subtract
from
0.5
Add to
0. 5
0.5
x
x
Add
C
Use
A=C–B
A B
"between x1 and x2 "
x1
x2
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
x1 x 2
34
Finding z Scores when
Given Probabilities
95%
5%
5% or 0.05
0.50
0.45
0
FIGURE 5-11
z
Finding the 95th Percentile
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Addison Wesley Longman
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Finding z Scores when
Given Probabilities
10%
90%
Bottom 10%
0.10 0.40
z
FIGURE 5-12
0
Finding the 10th Percentile
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
36
Finding a z - score when
given a probability
1. Identify the probability representing an
area 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.
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Addison Wesley Longman
37
Summary
 Two Distributions (continuous)
Uniform distribution
Standard Normal distribution
 Density Curve
 Finding Probabilities & Finding
Scores
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Addison Wesley Longman
38