Chap 5-1

Bab 5Distribusi Normal

Chap 5-2

Topik

distribusi normal

distribusi normal standar

Chap 5-3

Distribusi Probabilitas Kontinu

variabel random kontinuValues from interval of numbersAbsence of gaps

distribution probabilitas kontinuDistribution of continuous random variable

Most important continuous probability distribution

distribusi normal

Chap 5-4

Distribusi Normal

“Bell shaped”SymmetricalMean, median and mode are equalInterquartile rangeequals 1.33 σRandom variablehas infinite range

Mean Median Mode

X

f(X)

µ

Chap 5-5

Model Matematika

( )( )

( )

( )

212

2

12

: density of random variable 3.14159; 2.71828

: population mean: population standard deviation: value of random variable

Xf X e

f X Xe

X X

µσ

πσ

πµσ

2− −=

= =

−∞ < < ∞

Chap 5-6

Beberapa Distribusi Normal

distribusi normal dengan parameters

σ and µ, berbeda

Chap 5-7

Menentukan Nilai Probabilitas

Probability is the area under the curve!

c dX

f(X)

( ) ?P c X d≤ ≤ =

Chap 5-8

Tabel yang digunakan?

An infinite number of normal distributions means an infinite number of tables to look up!

Chap 5-9

Distribution Normal Standar

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

Shaded Area Exaggerated

0 1Z Zµ σ= =

Z = 0.120

Only One Table is Needed

Chap 5-10

Contoh6.2 5 0.12

10XZ µσ− −

= = =

Normal Distribution Standardized Normal Distribution

Shaded Area Exaggerated

10σ = 1Zσ =

5µ =6.2 X Z0Zµ =

0.12

Chap 5-11

Contoh( )2.9 7.1 .1664P X≤ ≤ =2.9 5 7.1 5.21 .21

10 10X XZ Zµ µσ σ− − − −

= = = − = = =

Normal Distribution Standardized Normal Distribution

Shaded Area Exaggerated

10σ = 1Zσ =

5µ =7.1 X Z0Zµ =

0.212.9 0.21−

.0832.0832

Chap 5-12

Contoh:( )2.9 7.1 .1664P X≤ ≤ =

(continued)

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)

Shaded Area Exaggerated

0 1Z Zµ σ= =

Z = 0.210

Chap 5-13

Contoh:( )2.9 7.1 .1664P X≤ ≤ =

(continued)

Z .00 .01

-03 .3821 .3783 .3745

.4207 .4168

-0.1.4602 .4562 .4522

0.0 .5000 .4960 .4920

.4168.02

-02 .4129

Cumulative Standardized Normal Distribution Table (Portion)

Shaded Area Exaggerated

0 1Z Zµ σ= =

Z = -0.210

Chap 5-14

Contoh:( )8 .3821P X ≥ =

8 5 .3010

XZ µσ− −

= = =

Normal Distribution Standardized Normal Distribution

Shaded Area Exaggerated

10σ = 1Zσ =

5µ =8 X Z0Zµ =

0.30

.3821

Chap 5-15

Contoh:( )8 .3821P X ≥ =

(continued)

Cumulative Standardized Normal Distribution Table (Portion) 0 1Z Zµ σ= =

Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.02

0.1 .5478

.6179

Shaded Area Exaggerated

Z = 0.300

Chap 5-16

Mengetahui nilai Zpada Probabilitas tertentu

Cumulative Standardized Normal Distribution Table

(Portion)What is Z Given Probability = 0.1217 ?

Z .00 0.2

0.0 .5000 .5040 .5080

0.1 .5398 .5438 .5478

0.2 .5793 .5832 .5871

.6179 .6255

.01

0.3

.6217

Shaded Area Exaggerated

.6217

0 1Z Zµ σ= =

.31Z =0

Chap 5-17

Nilai Xuntuk mengetahui Probabilitas

( )( )5 .30 10 8X Zµ σ= + = + =

Normal Distribution Standardized Normal Distribution

10σ = 1Zσ =

5µ = ? X Z0Zµ =0.30

.3821.1179

Chap 5-18

Assessing Normality(continued)

Normal Probability Plot for Normal Distribution

3060

90

-2 -1 0 1 2Z

X

Look for Straight Line!

Chap 5-19

Normal Probability Plot

Left-Skewed Right-Skewed

30

60

90

-2 -1 0 1 2Z

X30

60

90

-2 -1 0 1 2Z

X

Rectangular U-Shaped

30

60

90

-2 -1 0 1 2Z

X30

60

90

-2 -1 0 1 2Z

X

Chap 5-20

Larger sample size

Smaller sample size

P(X)

µ X

Chap 5-21

Populasi NormalPopulation Distribution

Sampling Distributions

Central Tendency

X50Xµ =

45X

nσ==

162.5X

nσ==

50µ =

10σ =

Xµ µ=

Variation

X nσσ =

Sampling with Replacement

Chap 5-22

Populasitidak Normal

Central Tendency

Variation

Sampling with Replacement

Population Distribution

Sampling Distributions

Xµ µ=

X nσσ =

X50Xµ =

45X

nσ==

301.8X

nσ==

50µ =

10σ =

Chap 5-23

Central Limit Theorem

As sample size gets large enough…

the sampling distribution becomes almost normal regardless of shape of population

X

Chap 5-24

Contoh:( )

8 =2 25

7.8 8.2 ?

n

P X

µ σ= =

< < =

( )

( )

7.8 8 8.2 87.8 8.22 / 25 2 / 25

.5 .5 .3830

X

X

XP X P

P Z

µσ

⎛ ⎞−− −< < = < <⎜ ⎟

⎝ ⎠= − < < =

Sampling Distribution Standardized Normal Distribution2 .4

25Xσ = = 1Zσ =

8Xµ =8.2 Z

0Zµ =0.57.8 0.5−

.1915

X