pemakaian distribusi normal

Student Lecture Notes 1

11

Penggunaan Penggunaan Distribusi NormalDistribusi Normal

1.1. Menjelaskkan banyak proses acak Menjelaskkan banyak proses acak yang kontinuyang kontinu

2.2. Bisa digunakan untuk mendekati Bisa digunakan untuk mendekati peluang perubah acak diskritpeluang perubah acak diskrit�� Example: BinomialExample: Binomial

3.3. Dasar dari semua statistik inferensia Dasar dari semua statistik inferensia klasikklasik

22

Normal DistributionNormal Distribution

1.1. ‘‘BellBell--ShapedShaped’’ & & SymmetricalSymmetrical

2.2. Mean, Median, Mean, Median, Mode Mode samasama

3.3. ‘‘Middle SpreadMiddle Spread’’adladl 1.33 1.33 σσ

4.4. Peubah Acak Peubah Acak mempunyai range tak mempunyai range tak hinggahingga

Mean Mean Median Median ModeMode

X

f(X)

33

Normal Distribution Normal Distribution Sifat yg pentingSifat yg penting

•• Hampir separo Hampir separo ““bobot/ bobot/ weightweight”” berada berada dibawah meandibawah mean (krn(krnsymmetri)symmetri)

•• 6868% % peluang berada peluang berada dlmdlm 1 standard 1 standard deviation deviation daridari mean mean

•• 9595% % peluang berada peluang berada dlmdlm 2 standard 2 standard deviationsdeviations

•• 9999% % peluang berada peluang berada dlmdlm 3 standard 3 standard deviations deviations

Mean Mean Median Median ModeMode

X

f(X)

σµ +

σµ − σµ +σµ 2− σµ 2+ σµ 3+σµ 3−

44

Probability Probability Density FunctionDensity Function

2

2

1

e2

1)(

−

−= σ

µ

πσ

x

xf

2

2

1

e2

1)(

−

−= σ

µ

πσ

x

xf

xx == Nilai Peubah acakNilai Peubah acak ((--∞∞ < < xx < < ∞∞))σσ == StandardStandard DeviationDeviation dari populasidari populasiππ == 3.141593.14159e = 2.71828e = 2.71828µµ == Mean Mean dari peubah acakdari peubah acak xx

Student Lecture Notes 2

55

NotasiNotasi

X X ~~ N(N(µµ,,σσ))

Peubah AcakPeubah Acak X mengikuti distribusi X mengikuti distribusi NormalNormal (N) (N) dengan meandengan mean µµ dandan standard standard deviation deviation σσ..

X X ~~ N(40,1)N(40,1)

X X ~~ N(10,5)N(10,5)

X X ~~ N(50,3)N(50,3)

66

Akibat dari Variasi Akibat dari Variasi ParameterParameter ((µµµµµµµµ & & σσσσσσσσ))

X

f(X)

CA

B

77

Normal Distribution Normal Distribution ProbabilityProbability

?)()( dxxfdxcPd

c∫=≤≤

c dx

f(x)

c dx

f(x)

Peluang Peluang dibawah dibawah kurva!kurva!

??

88

X

f(X)

X

f(X)

Tak hingga tabel NormalTak hingga tabel Normal

Tiap distribusi Tiap distribusi memerlukan satu tabel.memerlukan satu tabel.

Student Lecture Notes 3

99

Standardize theStandardize theNormal DistributionNormal Distribution

Xµµµµ

σσσσ

Xµµµµ

σσσσ

Hanya satu tabel!Hanya satu tabel!

Normal DistributionNormal Distribution

µµµµ = 0

σσσσ = 1

Zµµµµ = 0

σσσσ = 1

Z

ZX==== −−−− µµµµ

σσσσZ

X==== −−−− µµµµσσσσ Standardized

Normal DistributionStandardized

Normal Distribution

Z is N(0,1)Z is N(0,1)

1010

Contoh StandarisasiContoh Standarisasi

Xµµµµ= 5

σσσσ = 10

6.2 Xµµµµ= 5

σσσσ = 10

6.2

Normal DistributionNormal Distribution

ZX==== −−−− ==== −−−− ====µµµµ

σσσσ6 2 5

1012

..Z

X==== −−−− ==== −−−− ====µµµµσσσσ

6 2 510

12.

.

Zµµµµ= 0

σσσσ = 1

.12 Zµµµµ= 0

σσσσ = 1

.12

Standardized Normal Distribution

Standardized Normal Distribution

1111

Zµµµµ= 0

σσσσ = 1

.12 Zµµµµ= 0

σσσσ = 1

.12

Z .00 .01

0.0 .0000 .0040 .0080

.0398 .0438

0.2 .0793 .0832 .0871

0.3 .1179 .1217 .1255

Z .00 .01

0.0 .0000 .0040 .0080

.0398 .0438

0.2 .0793 .0832 .0871

0.3 .1179 .1217 .1255

Mendapatkan Mendapatkan PeluangnyaPeluangnya

.0478.0478.0478

.02.02

0.10.1 .0478

Standardized Normal Probability Table (Portion)Standardized Normal Standardized Normal Probability Table (Portion)Probability Table (Portion)

ProbabilitiesProbabilitiesProbabilities1212

ContohContohP(3.8P(3.8 ≤≤≤≤≤≤≤≤ XX ≤≤≤≤≤≤≤≤ 5)5)

Xµ µ µ µ = 5

σσσσ = 10

3.8 Xµ µ µ µ = 5

σσσσ = 10

3.8

Normal DistributionNormal Normal DistributionDistribution

ZX==== −−−− ==== −−−− ==== −−−−µµµµ

σσσσ3 8 5

1012

..Z

X==== −−−− ==== −−−− ==== −−−−µµµµσσσσ

3 8 510

12.

.

Zµµµµ = 0

σσσσ = 1

-.12 Zµµµµ = 0

σσσσ = 1

-.12

.0478.0478

Standardized Normal Distribution

Standardized Standardized Normal DistributionNormal Distribution

Shaded area exaggeratedShaded area exaggeratedShaded area exaggerated

Student Lecture Notes 4

1313

ContohContohP(2.9 P(2.9 ≤≤≤≤≤≤≤≤ XX ≤≤≤≤≤≤≤≤ 7.1) 7.1)

5

σσσσ = 10

2.9 7.1 X5

σσσσ = 10

2.9 7.1 X

Normal DistributionNormal Normal DistributionDistribution

ZX

ZX

====−−−− ====

−−−− ==== −−−−

====−−−− ====

−−−− ====

µµµµσσσσ

µµµµσσσσ

2 9 510

21

7 1 510

21

..

..

ZX

ZX

====−−−− ====

−−−− ==== −−−−

====−−−− ====

−−−− ====

µµµµσσσσ

µµµµσσσσ

2 9 510

21

7 1 510

21

..

..

0

σσσσ = 1

-.21 Z.210

σσσσ = 1

-.21 Z.21

.1664.1664.1664

.0832.0832.0832.0832

Standardized Normal Distribution

Standardized Standardized Normal DistributionNormal Distribution

Shaded area exaggeratedShaded area exaggeratedShaded area exaggerated 1414

Contoh Contoh P(P(XX ≥≥≥≥≥≥≥≥ 8)8)

Xµµµµ = 5

σσσσ = 10

8 Xµµµµ = 5

σσσσ = 10

8

Normal DistributionNormal Normal DistributionDistribution

Standardized Normal Distribution

Standardized Standardized Normal DistributionNormal Distribution

ZX==== −−−− ==== −−−− ====µµµµ

σσσσ8 510

30.ZX==== −−−− ==== −−−− ====µµµµ

σσσσ8 510

30.

Zµµµµ = 0

σσσσ = 1

.30 Zµµµµ = 0

σσσσ = 1

.30

.1179.1179.1179

.5000.5000.3821.3821.3821

Shaded area exaggeratedShaded area exaggeratedShaded area exaggerated

1515

Contoh Contoh P(7.1 P(7.1 ≤≤≤≤≤≤≤≤ XX ≤≤≤≤≤≤≤≤ 8)8)

µµµµ = 5

σσσσ = 10

87.1 Xµµµµ = 5

σσσσ = 10

87.1 X

Normal DistributionNormal Normal DistributionDistribution

ZX

ZX

====−−−− ====

−−−− ====

====−−−− ====

−−−− ====

µµµµσσσσ

µµµµσσσσ

7 1 510

21

8 510

30

..

.

ZX

ZX

====−−−− ====

−−−− ====

====−−−− ====

−−−− ====

µµµµσσσσ

µµµµσσσσ

7 1 510

21

8 510

30

..

.

µµµµ = 0

σσσσ = 1

.30 Z.21µµµµ = 0

σσσσ = 1

.30 Z.21

.0832.0832

.1179.1179 .0347.0347.0347

Standardized Normal Distribution

Standardized Standardized Normal DistributionNormal Distribution

Shaded area exaggeratedShaded area exaggeratedShaded area exaggerated