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Pertemuan 06
Sebaran Normal dan Sampling
1
Outline Materi:
• Peluang sebaran normal
• Sebaran rata-rata sampling
• Sebaran proporsi sampling
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 
X 
and a standard deviation s , Z 
follows
s
a standardized (normalized) normal distribution
with a mean 0 and a standard deviation 1.
s
f(Z) 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
Example
P  2.9  X  7.1  .1664(continued)
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
.02
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
Example
P  2.9  X  7.1  .1664(continued)
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
.02
Z  0
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
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
.02
(continued)
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
X    Zs  5  .3010  8
Z
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
Distribution Using PHStat
(continued)
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-andwhisker 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
26
s?
Assessing Normality
(continued)
• Observe the Distribution of the Data Set
– Do approximately 2/3 of the observations lie
 1 standard deviation?
between mean
– 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
27
with positive slope?
Assessing Normality
• Normal Probability Plot
(continued)
– 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
Sampling Distribution
• Sampling Distribution of the Mean
• The Central Limit Theorem
• Sampling Distribution of the Proportion
• Sampling from Finite Population
31
Why Study Sampling
Distributions
• Sample Statistics are Used to Estimate
Population Parameters

– E.g.,X  50 estimates the population mean
• Problem: Different Samples Provide
Different Estimates
– Large sample gives better estimate; large
sample costs more
– How good is the estimate?
• Approach to Solution: Theoretical Basis 32is
Sampling Distribution
• Theoretical Probability Distribution of a
Sample Statistic
• Sample Statistic is a Random Variable
– Sample mean, sample proportion
• Results from Taking All Possible Samples
of the Same Size
33
Developing Sampling
Distributions
• Suppose There is a Population …
• Population Size N=4
B
• Random Variable, X,
is Age of Individuals
• Values of X: 18, 20,
22, 24 Measured in
Years
C
D
A
34
Distributions
(continued)
Summary Measures for the Population Distribution
N

X
i 1
P(X)
i
.3
N
18  20  22  24

 21
4
N
s 
 X
i 1
i
N

.2
.1
0
2
 2.236
A
B
C
D
(18)
(20)
(22)
(24)
Uniform Distribution
X
35
Developing Sampling
Distributions
(continued)
All Possible Samples of Size n=2
1st
Obs
2nd Observation
18
20
22
24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
16 Sample Means
22 22,18 22,20 22,22 22,24
1st 2nd Observation
Obs 18 20 22 24
24 24,18 24,20 24,22 24,24
18 18 19 20 21
16 Samples Taken
with Replacement
20 19 20 21 22
22 20 21 22 23
24 21 22 23 3624
Developing Sampling
Distributions
(continued)
Sampling Distribution of All Sample Means
Sample Means
Distribution
16 Sample Means
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
.3
P X 
.2
.1
0
_
18 19
20 21 22 23
X
24
37
Developing Sampling
Distributions
(continued)
Summary Measures of Sampling Distribution
N
X 
X
i 1
N
i
18  19  19 

16
N
sX 
 X
i 1
i
 X 
 21
2
N
18  21  19  21
2

 24
16
2

  24  21
2
 1.58
38
Comparing the Population with Its
Sampling Distribution
Population
N=4
  21
P X 
s  2.236
Sample Means Distribution
n=2
 X  21
.3
.3
.2
.2
.1
.1
0
0
A
B
C
(18)
(20)
(22)
D X
P X 
s X  1.58
_
18 19
20 21 22 23
24
X
(24)
39
Properties of Summary Measures
• X  
– I.e., X is unbiased
• Standard Error (Standard Deviation) of the
s
Sampling Distribution X is Less Than the
Standard Error of Other Unbiased Estimators
• For Sampling with Replacement or without
Replacement
from Large or Infinite
s
s

X
Populations:
n
sX
40
Unbiasedness (
X  
)
f X 
Unbiased

Biased
X
X
41
Less Variability
Standard Error (Standard Deviation) of the
Sampling Distribution s X is Less Than the
Standard Error of Other Unbiased Estimators
f  X  Sampling
Distribution
of Median
Sampling
Distribution of
Mean

X
42
Effect of Large Sample
For sampling with replacement:
As n increases, s X decreases
f X 
Larger
sample size
Smaller
sample size

X
43
When the Population is
Normal
Population Distribution
Central Tendency
X  
Variation
sX 
s
n
s  10
  50
Sampling Distributions
n4
n  16
sX 5
s X  2.5
 X  50
X
44
When the Population is
Not Normal
Population Distribution
Central Tendency
X  
Variation
sX 
s
n
s  10
  50
Sampling Distributions
n4
n  30
sX 5
s X  1.8
 X  50
X
45
Central Limit Theorem
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost
Normal
Regardless
of Shape of
Population
X
46
How Large is Large Enough?
• For Most Distributions, n>30
• For Fairly Symmetric Distributions, n>15
• For Normal Distribution, the Sampling
Distribution of the Mean is Always
Normally Distributed Regardless of the
Sample Size
– This is a property of sampling from a normal
population distribution and is NOT a result of
the central limit theorem
47
Example:
 8
s =2
n  25
P  7.8  X  8.2   ?
 7.8  8 X   X 8.2  8 
P  7.8  X  8.2   P 



sX
2 / 25 
 2 / 25
 P  .5  Z  .5   .3830
Standardized
Normal Distribution
Sampling Distribution
2
sX 
 .4
25
sZ 1
.1915
7.8
8.2
X  8
X
0.5
Z  0
0.5
48
Z
Population Proportions
 p
• Categorical Variable
– E.g., Gender, Voted for Bush, College Degre
• Proportion of Population Having a
p

Characteristic
• Sample
Provides an Estimate
X Proportion
number of successes
pS 
–
n

sample size
• If Two Outcomes, X Has a Binomial
Distribution
49
Sampling Distribution of
Sample Proportion
• Approximated by
Normal Distribution
– np  5
n 1  p   5
– Mean:
•
p  p
f(ps)
.3
.2
.1
0
0
.2
.4
.6
8
ps
1
S
– Standard error:
•
Sampling Distribution
sp 
S
p 1  p 
n
p = population proportion
50
Standardizing Sampling
Distribution of Proportion
Z
pS   pS
sp
S
p 1  p 
n
Standardized
Normal Distribution
Sampling Distribution
sp

pS  p
sZ 1
S
p
S
pS
Z  0
Z
51
Example:
n  200
p  .4
P  pS  .43  ?

 p 
.43  .4
S
pS

P  pS  .43  P

 s pS
.4 1  .4 

200

Standardized
Normal Distribution
Sampling Distribution
sp


  P  Z  .87   .8078



sZ 1
S
 p .43
S
pS
0 .87
52
Z