k6 distribusi normal

Distribusi Normal

Topik :Pentingnya distribusi normalCiri-ciri distribusi normalPenggunaan tabel distribusi normalAplikasi Distribusi Normal

Which Table to Use?

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

Business Statistics, Chapter 6

]Chart5

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Pendahuluan:Fenomena : pengukuran yang berulang akan menghasilkan nilai yg berbeda-bedaNilai mana yg dianggap tepat?Karl Fred.Gauss : hasil pengukuran ulang yang sering terjadi adalah nilai rata-rata dan penyimpangan ke kanan dan ke kiri, makin jauh dari rata-rata, makin jarang terjadi dan pada akhirnya membentuk kurva yang sama dimana central tendensinya pada satu garis

Distribusi normal = Distribusi GaussMerupakan distribusi teoritis dari data kontinu

Penting ok:sesuai dgn distribusi frek.empirisuntuk pengambilan kesimpulan dr hasil sampel


GBS221, Business Statistics, Chapter 6

The mileage that an MX 100 tire gets is normally distributed around a mean of 47,900 miles.

Irwin/McGraw-Hill

The McGraw-Hill Companies, Inc., 1999

Areas Under the Normal Curve

Between:1.68.26%2.95.44%3.99.74%

Irwin/McGraw-Hill


7-10

Ciri-ciri distribusi normaldisusun dr var. random kontinu, uni modal, (kurvanya hanya memiliki satu puncak)simetris, btk lonceng, mean-median-mod terletak dalam satu titik, kurva normal dgn N tak terhingga,Grafik mendekati sumbu X pada penyimpangan 3 SD ke kanan dan ke kiri dari rata-rata. total daerah di bawah kurva nilai adalah satu.

The Normal Distribution Bell Shaped Symmetrical Mean, Median and Mode are Equal Random Variable hasInfinite RangeMean Median ModeXf(X)

Dalam praktek:ditulis dalam bentuk skor standar mean ()=0 dan standar deviasi () = 1Formula:A. cara ordinat dan B. cara luas

Mean Median Mode

A. tinggi kurva y untuk setiap nilai Xy = 1 e1/2 [(x) o- (m)]2 V2

B z = variasi nilai dalam unit standar deviasi sepanjang garis di bawah kurvaz = (X- )

ZtDistribusi0t (df = 5)Standard Normalt (df = 13)Bell-ShapedSymmetricFatter Tails

Irwin/McGraw-Hill



Between:1.68.26%2.95.44%3.99.74%

Irwin/McGraw-Hill


7-10

Tabel distribusi normal:

yi tabel utk menghitung luas daerah di bawah kurva pada setiap z

luas daerah di bawah kurva adalah 1, luas dr grs tengah pada titik nol ke kiri maupun ke kanan adalah 0,5

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

Shaded Area Exaggerated

Only One Table is Needed

Z = 0.12



Z ,00 ,01 ,02 ..,06 .,08,09

0,0 ,000 ,02390,11,6... ,4474..1,9 ,4750..2,5..,4951


Contoh 1:Luas daerah Z = 0 dan Z = 1?yi dengan mencari angka 1 pada kolom bawah dan angka 0,00 pada kolom atas : 0,3413


Z ,00 ,01 ,02 ..,06 .,08,09

0,0 ,000 ,02390,11,6... ,4474.1.0,34131,9 ,4750..2,5..,4951

Contoh 2 : Za = 1,96 dan Zb = -1,96. Luas daerah ?Cara : cari angka 1,9 pada kolom bawah dan 0,06 pada kolom atas Z---> luas daerah satu bagian(kanan) : 0,4750. Luas bagian kiri 0,4750 sehingga luas seluruh bagian : 0,95

Contoh 3 :Tentukan luas antara Za = 1,64 dan Zb = 1,96Za = 1,64 ---- 0,4495Zb = 1,96 ---- 0,4750Jadi luas antara : 0,4750-0,4495 = 0.0255

Soal :1 Tentukan luas daerah kurva, bila nilai Z = 0.82 Tentukan luas daerah kurva, bila nilai Z > 1, 3. Tentukan luas daerah kurva, bila nilai Z < 1,88

Irwin/McGraw-Hill


-4 -3 -2 -1 0 1 2 3 4

P(0

Irwin/McGraw-Hill


EXAMPLE 6

0 1 2 3 4

P(Z=1.88).5+.4699=.9699

Z=1.88

Irwin/McGraw-Hill


7-24

Distribusi NormalB Cara luas kurvaSetiap penyimpangan rata-rata dapat ditentukan persentase thd seluruh luas kurva.Penyimpangan 1 SD merupakan 68,26%penyimpangan 2 SD merupakan 95,5%penyimpangan 3 SD merupakan 99,7%Contoh : A : = 50, o = 20B : = 200 o = 10

Irwin/McGraw-Hill



Between:1.68.26%2.95.44%3.99.74%

Irwin/McGraw-Hill


7-10

Transformasi kurva distrib. normalDibentuk suatu kurva normal sebagai standar yang disebut kurva normal standarTransformasi rumus :Z = X- X = nilai var random = rata-rata distribusi = simpangan baku Z = nilai standar

Finding Probabilities

Probability is the area under the curve!

c

d

X

f(X)


z - Score

Number of standard deviations an observation resides from the mean.

Standardizing Example

Normal Distribution

Standardized Normal Distribution

Shaded Area Exaggerated


Aplikasi distribusi normalContoh : diketahui mean (rata-rata) TB laki-laki atau = 160 cm dan standar deviasi = 5 cm. Jlh populasi=100 Pertanyaan: a. Berapa proporsi laki-laki dengan tinggi badan < 170 cm? b. Berapa proporsi laki-laki dengan TB 165-170? c. Berapa nilai Z dan proporsi pada laki-laki dengan tinggi badan antara 155 cm dan 167 d. berapa jlh laki-laki dengan TB .155- 167

Empirical Rule

The distance between X and can be measured in miles or standard deviations.Z= the number of standard deviations that X is from


GBS221, Business Statistics, Chapter 6

Notice that the 68%, 95%, and 99.7% are dependent only on the number of standard Deviations we are from the mean.

That means, for any normal probability distribution the above relationship holds.

All we need to know is the number of standard deviations we are from the mean.

Distribusi samplingDistribusi dari mean-mean sampel yang diambil secara berulang kali dari suatu populasi.

Ukuran untuk sampel dan populasi

sampelpopulasinilaistatistikparametermeanXMiuStandar DevsThoJumlah unitnN

Distribusi samplingPopulasiX1,X2,..Xn(mean) / (sd)

sampel 1 sampel 2 sampel 3xi.xn xi..xn xixnx1 x2 x3 X (rata-rata)Distribusi sampling = x (rata-rata) SE= / n

Distribusi samplingDistribusi sampling = central limit teorem1.Bila sampel random dengan n (jlh sampel) diambil dr suatu populasi, maka mean dari distribusi sampling = mean populasi (), standar deviasi distribusi sampling = / n dikenal SE (standar error)

Distribusi sampling

2.Bila populasi berdistribusi normal, maka distribusi sampling mean (X) akan berdistribusi normal shg berlaku sifat Z = (X- )/ SE

Distribusi sampling3. Walaupun populasi berdistribusi sembarang (tidak normal), kalau diambil sampel berulang kali secara random, maka distribusi harga mean akan berbentuk distribusi normal. Contoh:Populasi penderita penyakit X jlh 5 orangmasa inkubasi rata-rata( )= 6 hari, standar deviasi () = 3,29 haridiambil sampel dengan besar n=2hasil rata-rata (X) = 6, SE = / n = 2,32

Distribusi sampling

Contoh:Populasi penderita penyakit X jlh 5 orangmasa inkubasi rata-rata( )= 6 hari, standar deviasi () = 3,29 haridiambil sampel dengan besar n=2hasil rata-rata (X) = 6, SE = / n = 2,32

Latihan Diketahui bahwa berat bayi lahir aterm terletak antara 2500 gr dan 4000 gr. Jumlah bayi yang dilahirkan adalah 1000, rata-rata 3000 gr dan Sd 500 grA. Berapa persen bayio dengan BBL > 4500B. Berapa jumlah bayi dengan BBL antara 2500 dan 3500 gr

*

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