1 pertemuan 11 peubah acak normal matakuliah: i0134-metode statistika tahun: 2007
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Pertemuan 11Peubah Acak Normal
Matakuliah : I0134-Metode Statistika
Tahun : 2007
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Outline Materi:• Peluang sebaran normal
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Basic Business Statistics (9th Edition)
The Normal Distribution and Other Continuous Distributions
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Peluang sebaran normal
• The Normal Distribution
• The Standardized Normal Distribution
• Evaluating the Normality Assumption
• The Uniform Distribution
• The Exponential Distribution
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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
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The Normal Distribution• “Bell Shaped”• Symmetrical• Mean, Median and
Mode are Equal• Interquartile Range
Equals 1.33 • Random Variable
Has Infinite Range Mean Median Mode
X
f(X)
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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
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Many Normal Distributions
Varying the Parameters and , We Obtain Different Normal Distributions
There are an Infinite Number of Normal Distributions
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The Standardized Normal 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.
XZ
X
f(X)
Z
0Z
1Z
f(Z)
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Finding Probabilities
Probability is the area under the curve!
c dX
f(X)
?P c X d
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Which Table to Use?
Infinitely Many Normal Distributions Means Infinitely Many Tables to Look
Up!
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Solution: The Cumulative Standardized 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 Normal Distribution Table (Portion)
Probabilities
Only One Table is Needed
0 1Z Z
Z = 0.12
0
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Standardizing Example
6.2 50.12
10
XZ
Normal Distribution
Standardized Normal
Distribution10 1Z
5 6.2 X Z
0Z 0.12
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Example
Normal Distribution
Standardized Normal
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
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Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.5832.02
0.1 .5478
Cumulative Standardized Normal Distribution Table (Portion)
0 1Z Z
Z = 0.21
Example 2.9 7.1 .1664P X
(continued)
0
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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 Normal Distribution Table (Portion)
0 1Z Z
Z = -0.21
Example 2.9 7.1 .1664P X
(continued)
0
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Normal Distribution in PHStat
• PHStat | Probability & Prob. Distributions | Normal …
• Example in Excel Spreadsheet
Microsoft Excel Worksheet
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Example :
8 .3821P X
Normal Distribution
Standardized Normal
Distribution10 1Z
5 8 X Z0Z
0.30
8 5.30
10
XZ
.3821
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Example:
Example: 8 .3821P X
(continued)
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 Normal Distribution Table (Portion)
0 1Z Z
Z = 0.30
0
20
.6217
Finding Z Values for Known Probabilities
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 Standardized Normal Distribution Table
(Portion)
What is Z Given Probability = 0.6217 ?
.6217
0 1Z Z
.31Z 0
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Recovering X Values for Known Probabilities
5 .30 10 8X Z
Normal Distribution
Standardized Normal
Distribution10 1Z
5 ? X Z0Z 0.30
.3821.6179
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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?
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
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What percentage of students scored between 65 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 Distribution Using PHStat
(continued)
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73
Only 5% of the students taking the test scored 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.
1.6450
X
Z? =86.16
(continued)
More Examples of Normal Distribution Using PHStat
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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
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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 ?– Is the range approximately 6 ?
(continued)
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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)
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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 Standardized Normal Quantile Values on the Horizontal Axis
– Evaluate the Plot for Evidence of Linearity
(continued)
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Assessing Normality
Normal Probability Plot for Normal Distribution
Look for Straight Line!
30
60
90
-2 -1 0 1 2
Z
X
(continued)
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Normal Probability Plot
Left-Skewed Right-Skewed
Rectangular U-Shaped
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X