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DistribusiNormal
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
2
CHAPTER TOPICS
The Normal Distribution
The Standardized Normal Distribution
Evaluating the Normality Assumption
The Uniform Distribution
The Exponential Distribution
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
3
CONTINUOUS PROBABILITYDISTRIBUTIONS
Continuous Random Variable Values from interval of numbers Absence of gaps
Continuous Probability Distribution Distribution of continuous random variable
Most Important Continuous ProbabilityDistribution The normal distribution
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
4
THE NORMAL DISTRIBUTION
“Bell Shaped” Symmetrical Mean, Median and
Mode are Equal Interquartile Range
Equals 1.33 σ Random Variable
Has Infinite Range
MeanMedianMode
X
f(X)
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
5
THE MATHEMATICAL MODEL
2(1/ 2) /1
2
: density of random variable
3.14159; 2.71828
: population mean
: population standard deviation
: value of random variable
Xf X e
f X X
e
X X
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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MANY NORMALDISTRIBUTIONS
Varying the Parameters and , We ObtainDifferent Normal Distributions
There are an Infinite Number of Normal Distributions
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
7
THE STANDARDIZEDNORMAL DISTRIBUTION
When X is normally distributed with a mean and a
standard deviation , follows a standardized
(normalized) normal distribution with a mean 0 and a
standard deviation 1.
X
f(X)
Z
0Z
1Z
f(Z)
XZ
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
8
FINDING PROBABILITIES
Probability isthe area underthe curve!
c dX
f(X)
?P c X d
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
9
WHICH TABLE TO USE?
Infinitely Many Normal DistributionsMeans Infinitely Many Tables to Look Up!
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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SOLUTION: THE CUMULATIVESTANDARDIZED NORMAL DISTRIBUTION
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.5478.02
0.1 .5478
Cumulative Standardized NormalDistribution Table (Portion)
Probabilities
Only One Table is Needed
0 1Z Z
Z = 0.120
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
11
STANDARDIZING EXAMPLE6.2 5
0.1210
XZ
Normal Distribution StandardizedNormal Distribution
10 1Z
5 6.2 X Z
0Z 0.12
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
12
EXAMPLE:
Normal Distribution StandardizedNormal Distribution10 1Z
5 7 .1 X Z0Z
0.21
2 .9 5 7 .1 5.21 .21
10 10
X XZ Z
2 .9 0 .21
.0832
2.9 7 .1 .1664P X
.0832
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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EXAMPLE:
Z .00 .010.0 .5000 .5040 .5080
.5398 .54380.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.5832.02
0.1 .5478
Cumulative Standardized NormalDistribution Table (Portion) 0 1Z Z
Z = 0.21
2.9 7.1 .1664P X (continued)
0
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
14
EXAMPLE:
Z .00 .01
-0.3 .3821 .3783 .3745
.4207 .4168
-0.1 .4602 .4562 .4522
0.0 .5000 .4960 .4920
.4168.02
-0.2 .4129
Cumulative Standardized NormalDistribution Table (Portion) 0 1Z Z
Z = -0.21
2.9 7.1 .1664P X (continued)
0
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
15
NORMAL DISTRIBUTIONIN PHSTAT
PHStat | Probability & Prob. Distributions |Normal …
Example in Excel Spreadsheet
Microsoft ExcelWorksheet
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
16
EXAMPLE:
5
8 .3821P X
Normal Distribution StandardizedNormal Distribution
10 1Z
8 X Z0Z 0.30
8 5.30
10
XZ
.3821
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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EXAMPLE: (continued)
8 .3821P X
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.6179.02
0.1 .5478
Cumulative Standardized NormalDistribution Table (Portion) 0 1Z Z
Z = 0.300
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
18
FINDING Z VALUES FORKNOWN PROBABILITIES
.6217
Z .00 0.2
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
.6179 .6255
.01
0.3
Cumulative StandardizedNormal Distribution Table
(Portion)What is Z GivenProbability = 0.6217 ?
.6217
0 1Z Z
.31Z 0
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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RECOVERING X VALUES FORKNOWN PROBABILITIES
5 .30 10 8X Z
Normal Distribution StandardizedNormal Distribution
10 1Z
5 ? X Z0Z 0.30
.3821.6179
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
20
MORE EXAMPLES OF NORMAL DISTRIBUTIONUSING PHSTAT
A set of final exam grades was found to be normallydistributed with a mean of 73 and a standard deviation of 8.
What is the probability of getting a grade no higher than 91on this exam?
273,8X N 91 ?P X Mean 73Standard Deviation 8
X Value 91Z Value 2.25P(X<=91) 0.9877756
Probability for X <=
2.250
X
Z91
8
73
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
21
What percentage of students scored between65 and 89?
From X Value 65To X Value 89Z Value for 65 -1Z Value for 89 2P(X<=65) 0.1587P(X<=89) 0.9772P(65<=X<=89) 0.8186
Probability for a Range
273,8X N 65 89 ?P X
20
X
Z8965
-1
73
MORE EXAMPLES OF NORMAL DISTRIBUTIONUSING PHSTAT
(continued)
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
221.645
73
Only 5% of the students taking the testscored higher than what grade? 273,8X N ? .05P X
Cumulative Percentage 95.00%Z Value 1.644853X Value 86.15882
Find X and Z Given Cum. Pctage.
0
X
Z? =86.16
(continued)
MORE EXAMPLES OF NORMALDISTRIBUTION USING PHSTAT
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
23
ASSESSING NORMALITY
Not All Continuous Random Variables areNormally Distributed
It is Important to Evaluate How Well the DataSet Seems to Be Adequately Approximated bya Normal Distribution
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
24
ASSESSING NORMALITY
Construct Charts For small- or moderate-sized data sets, do the stem-and-
leaf display and box-and-whisker plot look symmetric? For large data sets, does the histogram or polygon appear
bell-shaped?
Compute Descriptive Summary Measures Do the mean, median and mode have similar values? Is the interquartile range approximately 1.33 σ? Is the range approximately 6 σ?
(continued)
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
25
ASSESSING NORMALITY
Observe the Distribution of the Data Set Do approximately 2/3 of the observations lie
between mean 1 standard deviation? Do approximately 4/5 of the observations lie
between mean 1.28 standard deviations? Do approximately 19/20 of the observations lie
between mean 2 standard deviations? Evaluate Normal Probability Plot Do the points lie on or close to a straight line
with positive slope?
(continued)
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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ASSESSING NORMALITY
Normal Probability Plot Arrange Data into Ordered Array Find Corresponding Standardized Normal
Quantile Values Plot the Pairs of Points with Observed Data
Values on the Vertical Axis and the StandardizedNormal Quantile Values on the Horizontal Axis
Evaluate the Plot for Evidence of Linearity
(continued)
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
27
ASSESSING NORMALITY
Normal Probability Plot for NormalDistribution
Look for Straight Line!
306090
-2 -1 0 1 2Z
X
(continued)
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
28
NORMAL PROBABILITY PLOT
Left-Skewed Right-Skewed
Rectangular U-Shaped
306090
-2 -1 0 1 2Z
X306090
-2 -1 0 1 2Z
X
306090
-2 -1 0 1 2Z
X306090
-2 -1 0 1 2Z
X
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
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OBTAINING NORMAL PROBABILITYPLOT IN PHSTAT
PHStat | Probability & Prob. Distributions |Normal Probability Plot
Enter the range of the cells that contain thedata in the Variable Cell Range window
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
30
THE UNIFORMDISTRIBUTION
Properties: The probability of occurrence of a value is
equally likely to occur anywhere in the rangebetween the smallest value a and the largestvalue b
Also called the rectangular distribution
2
a b
22
12
b a
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
31
THE UNIFORMDISTRIBUTION
The Probability Density Function
Application: Selection of random numbers E.g., A wooden wheel is spun on a horizontal
surface and allowed to come to rest. What is theprobability that a mark on the wheel will point tosomewhere between the North and the East?
(continued)
1
iff X a X bb a
900 90 0.25
360P X
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
32
EXPONENTIALDISTRIBUTIONS
arrival tim e 1
: any value o f con tinuous random variab le
: the popu lation average num ber o f
arrivals per un it o f tim e
1 / : average tim e betw een arrivals
2 .71828
XP X e
X
e
E.g., Drivers arriving at a toll bridge;customers arriving at an ATM machine
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
33
EXPONENTIALDISTRIBUTIONS
Describes Time or Distance between Events Used for queues
Density Function
Parameters
(continued)
f(X)
X
= 0.5
= 2.0
1 x
f x e
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
34
EXAMPLE
E.g., Customers arrive at the checkout line of asupermarket at the rate of 30 per hour. What isthe probability that the arrival time betweenconsecutive customers will be greater than 5minutes?
3 0 5 / 6 0
3 0 5 / 6 0 h o u rs
arriva l tim e > 1 arriva l tim e
1 1
.0 8 2 1
X
P X P X
e
STATISTIK & PROBABILITASCopyright © 2017 By. Ir. Arthur Daniel Limantara, MM, MT.
35
EXPONENTIAL DISTRIBUTIONIN PHSTAT
PHStat | Probability & Prob. Distributions |Exponential
Example in Excel Spreadsheet
Microsoft ExcelWorksheet
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