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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

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: 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

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.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

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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