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
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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
Business Statistics, Chapter 6
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
The McGraw-Hill Companies, Inc., 1999
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
The McGraw-Hill Companies, Inc., 1999
Areas Under the Normal Curve
Between:1.68.26%2.95.44%3.99.74%
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
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
Business Statistics, Chapter 6
Tabel distribusi normal:
Z ,00 ,01 ,02 ..,06 .,08,09
0,0 ,000 ,02390,11,6... ,4474..1,9 ,4750..2,5..,4951
Tabel distribusi normal:
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
Tabel distribusi normal:
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
The McGraw-Hill Companies, Inc., 1999
-4 -3 -2 -1 0 1 2 3 4
P(0
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
-4 -3 -2 -1 0 1 2 3 4
P(0
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
EXAMPLE 6
0 1 2 3 4
P(Z=1.88).5+.4699=.9699
Z=1.88
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
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
The McGraw-Hill Companies, Inc., 1999
Areas Under the Normal Curve
Between:1.68.26%2.95.44%3.99.74%
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
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)
Business Statistics, Chapter 6
z - Score
Number of standard deviations an observation resides from the mean.
Standardizing Example
Normal Distribution
Standardized Normal Distribution
Shaded Area Exaggerated
Business Statistics, Chapter 6
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
Business Statistics, Chapter 6
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
x
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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