Download Normal Probability Distributions

ELEMENTARY
Chapter 5
STATISTICS
Normal Probability Distributions
MARIO F. TRIOLA
EIGHTH
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
EDITION1
Chapter 5
Normal Probability Distributions
5-1
Overview
5-2
The Standard Normal Distribution
5-3
Normal Distributions: Finding Probabilities
5-4
Normal Distributions: Finding Values
5-5
The Central Limit Theorem
5-6
Normal Distribution as Approximation to
Binomial Distribution
5-7
Determining Normality
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
5-1
Overview
 Continuous random variable
 Normal distribution
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
5-1
Overview
 Continuous random variable
 Normal distribution
Curve is bell shaped
and symmetric
Figure 5-1
µ
Score
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
5-1
Overview
 Continuous random variable
 Normal distribution
Curve is bell shaped
and symmetric
Figure 5-1
µ
Score
Formula 5-1
y=
e
1
2

x-µ
2
(  )
2p
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
5-2
The Standard Normal
Distribution
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Definitions
 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.
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Definitions
 Density Curve (or probability
density
function)
the graph of a continuous
probability distribution
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Definitions
 Density Curve (or probability
density
function)
the graph of a continuous
probability distribution
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.
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Because the total area under
the density curve is equal to 1,
there is a correspondence
between area and probability.
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Times in First or Last Half Hours
Figure 5-3
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Heights of Adult Men and Women
Women:
µ = 63.6
 = 2.5
Figure 5-4
Men:
µ = 69.0
 = 2.8
63.6
69.0
Height (inches)
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Definition
Standard Normal Deviation
a normal probability distribution that has a
mean of 0 and a standard deviation of 1
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Definition
Standard Normal Deviation
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
2
3
0
z = 1.58
Score (z )
Figure 5-5
Figure 5-6
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Table A-2
Standard Normal Distribution
=1
µ=0
0
x
z
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
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
*
*
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
To find:
z Score
the 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
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
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.
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18
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.
P ( 0 < x < 1.58 ) =
0
1.58
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
*
*
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
20
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.
Area = 0.4429
P ( 0 < x < 1.58 ) = 0.4429
0
1.58
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
21
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.
Area = 0.4429
P ( 0 < x < 1.58 ) = 0.4429
0
1.58
The probability that the chosen
thermometer will measure freezing water
between 0 and 1.58 degrees is 0.4429.
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
22
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.
Area = 0.4429
P ( 0 < x < 1.58 ) = 0.4429
0
1.58
There is 44.29% of the thermometers with
readings between 0 and 1.58 degrees.
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
23
Using Symmetry to Find the Area
to the Left of the Mean
Because of symmetry, these areas are equal.
Figure 5-7
(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 probability)
can never be negative.
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
24
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.
Area = 0.4925
P ( -2.43 < x < 0 ) = 0.4925
-2.43
0
The probability that the chosen thermometer
will measure freezing water between -2.43
and 0 degrees is 0.4925.
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
25
The Empirical Rule
Standard Normal Distribution: µ = 0 and  = 1
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
26
The Empirical Rule
Standard Normal Distribution: µ = 0 and  = 1
68% within
1 standard deviation
34%
x-s
34%
x
x+s
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
27
The Empirical Rule
Standard Normal Distribution: µ = 0 and  = 1
95% within
2 standard deviations
68% within
1 standard deviation
34%
34%
13.5%
x - 2s
13.5%
x-s
x
x+s
x + 2s
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
28
The Empirical Rule
Standard Normal Distribution: µ = 0 and  = 1
99.7% of data are within 3 standard deviations of the mean
95% within
2 standard deviations
68% within
1 standard deviation
34%
34%
2.4%
2.4%
0.1%
0.1%
13.5%
x - 3s
x - 2s
13.5%
x-s
x
x+s
x + 2s
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
x + 3s
29
Probability of Half of a Distribution
0.5
0
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
30
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
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
31
Finding the Area Between z = 1.20 and z = 2.30
0.4893 (from Table A-2 with z = 2.30)
Area A is 0.4893 - 0.3849 =
0.1044
0.3849
A
0
z = 1.20 z = 2.30
Figure 5-9
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
32
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
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
33
Interpreting Area Correctly
Figure 5-10
‘greater than
‘at least
x’
x’
‘more than
Subtract
from
0.5
Add to
0.5
x’
‘not less than
x’
0.5
x
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
x
34
Interpreting Area Correctly
Figure 5-10
‘greater than
‘at least
x’
Add to
0.5
x’
‘more than
Subtract
from
0.5
x’
‘not less than
x’
0.5
x
Add to
0.5
x
‘less than
‘at most
x’
x’
‘no more than
x’
‘not greater than
Subtract
from
0.5
x’
0.5
x
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
x
35
Interpreting Area Correctly
Figure 5-10
‘greater than
‘at least
x’
Add to
0.5
x’
‘more than
Subtract
from
0.5
x’
‘not less than
x’
0.5
x
Add to
0.5
x
‘less than
‘at most
x’
x’
‘no more than
x’
‘not greater than
Subtract
from
0.5
x’
0.5
x
x
Add
C
‘between
x1
and
Use
A=C-B
x 2’
A
x1
x2
B
x1 x2
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
36
Finding a z - score when given a probability
Using 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.
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
37
Finding z Scores when Given Probabilities
95%
5%
5% or 0.05
0.45
0.50
z
0
( z score will be positive )
FIGURE 5-11
Finding the 95th Percentile
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
38
Finding z Scores when Given Probabilities
95%
5%
5% or 0.05
0.45
0.50
0
1.645
(z score will be positive)
FIGURE 5-11
Finding the 95th Percentile
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
39
Finding z Scores when Given Probabilities
90%
10%
Bottom 10%
0.10
0.40
z
0
(z score will be negative)
FIGURE 5-12
Finding the 10th Percentile
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
40
Finding z Scores when Given Probabilities
90%
10%
Bottom 10%
0.10
0.40
-1.28
0
(z score will be negative)
FIGURE 5-12
Finding the 10th Percentile
Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
41