Matakuliah
Tahun
: I0134-Metode Statistika
: 2007
Pertemuan 11
Peubah Acak Normal
1
Outline Materi:
• Peluang sebaran normal
2
Basic Business Statistics
(9th Edition)
The Normal Distribution and Other
Continuous Distributions
3
Peluang sebaran normal
• The Normal Distribution
• The Standardized Normal Distribution
• Evaluating the Normality Assumption
• The Uniform Distribution
• The Exponential Distribution
4
Continuous Probability Distributions
• Continuous Random Variable
– Values from interval of numbers
– Absence of gaps
• Continuous Probability Distribution
– Distribution of continuous random variable
• Most Important Continuous Probability Distribution
– The normal distribution
5
The Normal Distribution
• “Bell Shaped”
• Symmetrical
• Mean, Median and
Mode are Equal
• Interquartile Range
Equals 1.33 s
• Random Variable
Has Infinite Range
f(X)

X
Mean
Median
Mode
6
The Mathematical Model
2
1
 (1/ 2)  X    / s 
f X  
e
2s
f  X  : density of random variable X
  3.14159;
e  2.71828
 : population mean
s : population standard deviation
X : value of random variable    X   
7
Many Normal Distributions
There are an Infinite Number of Normal Distributions
Varying the Parameters s and , We Obtain
Different Normal Distributions
8
The Standardized Normal Distribution
When X is normally distributed with a mean
deviation
and a standard

follows a standardizedX
(normalized)
normal

,
s Z
distribution with a mean 0 and a standard deviation
s 1.
f(Z)
s
f(X)
sZ 1

Z  0
X
Z
9
Finding Probabilities
Probability is
the area under
the curve!
P c  X  d   ?
f(X)
c
d
X
10
Which Table to Use?
Infinitely Many Normal Distributions
Means Infinitely Many Tables to Look Up!
11
Solution: The Cumulative
Standardized Normal Distribution
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
sZ 1
.02
.5478
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
Probabilities
0.3 .6179 .6217 .6255
0
Z = 0.12
Only One Table is Needed
12
Standardizing Example
Z
X 
s
6.2  5

 0.12
10
Standardized
Normal Distribution
Normal Distribution
s  10
sZ 1
6.2
 5
X
0.12
Z  0
Z
13
Example
P  2.9  X  7.1  .1664
Z
X 
s
2.9  5

 .21
10
Z
X 
s
7.1  5

 .21
10
Standardized
Normal Distribution
Normal Distribution
s  10
.0832
sZ 1
.0832
2.9 7.1
 5
X
0.21 0.21
Z  0
Z
14
P  2.9  X  7.1  .1664
Example
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
.02
(continued)
sZ 1
.5832
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
0
Z = 0.21
15
P  2.9  X  7.1  .1664
Example
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
.02
Z  0
(continued)
sZ 1
.4168
-0.3 .3821 .3783 .3745
-0.2 .4207 .4168 .4129
-0.1 .4602 .4562 .4522
0.0 .5000 .4960 .4920
0
Z = -0.21
16
Normal Distribution in PHStat
• PHStat | Probability & Prob. Distributions | Normal …
• Example in Excel Spreadsheet
17
Example :
P  X  8  .3821
Z
X 
s
85

 .30
10
Standardized
Normal Distribution
Normal Distribution
s  10
sZ 1
.3821
 5
8
X
0.30
Z  0
Z
18
Example:
P  X  8  .3821
(continued)
Example:
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
.02
sZ 1
.6179
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
0
Z = 0.30
19
Finding Z Values for Known Probabilities
What is Z Given
Probability = 0.6217 ?
Z  0
sZ 1
Cumulative Standardized
Normal Distribution Table
(Portion)
Z
.00
.01
0.2
0.0 .5000 .5040 .5080
.6217
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0
Z  .31
0.3 .6179 .6217 .6255
20
Recovering X Values for Known Probabilities
Standardized
Normal Distribution
Normal Distribution
s  10
sZ 1
.6179
.3821
 5
?
X
Z  0
0.30
Z
X    Zs  5  .3010  8
21
More Examples of Normal Distribution Using PHStat
A set of final exam grades was found to be normally
distributed with a mean of 73 and a standard deviation of 8.
What is the probability of getting a grade no higher than 91
on this exam?
X
N  73,8
2

Mean
Standard Deviation
P  X  91  ?
s 8
73
8
Probability for X <=
X Value
91
Z Value
2.25
P(X<=91)
0.9877756
X
  73 91
0
2.25
Z
22
(continued)
More Examples of Normal Distribution Using PHStat
What percentage of students scored between
65 and 89?
X
N  73,82 
P  65  X  89  ?
Probability for a Range
From X Value
65
To X Value
89
Z Value for 65
-1
Z Value for 89
2
P(X<=65)
0.1587
P(X<=89)
0.9772
P(65<=X<=89)
0.8186
X
65
  73 89
-1 0
2
Z
23
More Examples of Normal Distribution
Using PHStat
(continued)
Only 5% of the students taking the test
scored higher than what grade?
X
N  73,8
2

P  ?  X   .05
Find X and Z Given Cum. Pctage.
Cumulative Percentage
95.00%
Z Value
1.644853
X Value
86.15882
X
  73 ? =86.16
0
1.645
24
Z
Assessing Normality
• Not All Continuous Random Variables are Normally
Distributed
• It is Important to Evaluate How Well the Data Set Seems
to Be Adequately Approximated by a Normal Distribution
25
Assessing Normality
(continued)
• 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 bellshaped?
• Compute Descriptive Summary Measures
– Do the mean, median and mode have similar values?
– Is the interquartile range approximately 1.33 s?
– Is the range approximately 6 s?
26
Assessing Normality
(continued)
• 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?
27
Assessing Normality
(continued)
• 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 Standardized Normal Quantile Values on
the Horizontal Axis
– Evaluate the Plot for Evidence of Linearity
28
Assessing Normality
(continued)
Normal Probability Plot for Normal
Distribution
90
X 60
Z
30
-2 -1 0 1 2
Look for Straight Line!
29
Normal Probability Plot
Left-Skewed
Right-Skewed
90
90
X 60
X 60
Z
30
-2 -1 0 1 2
-2 -1 0 1 2
Rectangular
U-Shaped
90
90
X 60
X 60
Z
30
-2 -1 0 1 2
Z
30
Z
30
-2 -1 0 1 2
30