Laboratorium Statistika dan Komputasi Matematika

Fakultas Matematika dan Ilmu Pengetahuan Alam

:: Praktikum Statistika menggunakan R ::

05. Uji Hipotesis

Uji Hipotesis

Kelompok Keilmuan Statistika

MA2181 Analisis Data / MA2081 Statistika Dasar / MA2082 Biostatistika

Copyright 2020 © KK Statistika, FMIPA – ITB

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Mempelajari pengujian hipotesis terhadap distribusi normal.

Melakukan uji hipotesis pada beberapa contoh masalah.

TUJUAN

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2

Copyright 2020 © KK Statistika, FMIPA – ITB

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Uji Hipotesis

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Langkah Uji Hipotesis

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Uji Hipotesis Rataan 1 Populasi

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Uji Hipotesis Rataan 1 Populasi

#Cara Manual

#Input

x = read.csv(“nama file.csv”) #data

xbar = mean(x) #mean sampel

mu0 #nilai hipotesis

sigma #standar deviasi populasi

n = length(x) #banyak observasi

alpha = 0.05 #taraf signifikansi

​

#Statistika uji (bandingkan z hitung dengan z tabel)

z = (xbar-mu0)/(sigma/sqrt(n)) #z hitung

z.lower = qnorm(alpha) #z tabel eka arah

z.upper = qnorm(1-alpha) #z tabel eka arah

z.half.alpha = qnorm(1-alpha/2) #z tabel dwi arah

z.twosided = c(-z.half.alpha, z.half.alpha)

​

#p-value (bandingkan p-value dengan alpha)

pval.lower = pnorm(z) #eka arah

pval.upper = pnorm(z, lower.tail = FALSE) #eka arah

pval.twosided = 2*pnorm(z) #dwi arah

#Cara Otomatis

library(TeachingDemos)

z.test(x, mu=mu0, sd=sigma,

alternative = c(“two.sided”,”less”,”greater”),

conf.level = 0.95)

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Uji Hipotesis Rataan 1 Populasi

#Cara Manual

#Input

x #data

xbar = mean(x) #mean sampel

mu0 #nilai hipotesis

s = sd(x) #standar deviasi sampel

n = length(x) #banyak observasi

alpha = 0.05 #taraf signifikansi

​

#Statistika uji (bandingkan t hitung dengan t tabel)

t = (xbar-mu0)/(s/sqrt(n)) #t hitung

t.lower = qt(alpha,df = n-1) #t tabel eka arah

t.upper = qt(1-alpha, df = n-1) #t tabel eka arah

t.half.alpha = qt(1-alpha/2, df = n-1) #t tabel dwi arah

t.twosided = c(-t.half.alpha, t.half.alpha)

​

#p-value (bandingkan p-value dengan alpha)

pval.lower = pt(t, df=n-1) #eka arah

pval.upper = pt(t, df=n-1, lower.tail = FALSE) #eka arah

pval.twosided = 2*pt(t, df=n-1) #dwi arah

#Cara Otomatis

t.test(x, mu=mu0, alternative = c(“two.sided”,”less”,”greater”), conf.level = 0.95)

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Contoh Soal 1

Copyright 2020 © KK Statistika, FMIPA – ITB

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Prosedur R Soal 1

setwd(“Alamat folder")

library(readxl)

​

#Input

x = read_excel("DATA UJI HIPOTESIS.xlsx", sheet = "contoh diameter logam")

x = as.numeric(x$`diameter potongan logam`)

xbar = mean(x) #mean sampel

mu0 = 1.09 #nilai hipotesis

s = sd(x) #standar deviasi sampel

n = length(x) #banyak observasi

alpha = 0.05 #taraf signifikansi

​

#Cara Manual (bandingkan t hitung dengan t tabel)

(t = (xbar-mu0)/(s/sqrt(n))) #t hitung

(t.lower = qt(alpha,df = n-1)) #t tabel eka arah

​

#p-value (bandingkan p-value dengan alpha)

(pval.lower = pt(t, df=n-1)) #eka arah

​

#Cara Otomatis

t.test(x, mu=mu0, alternative = "less", conf.level = 0.95)

Editor

> #Cara Manual (bandingkan t hitung dengan t tabel)

> (t = (xbar-mu0)/(s/sqrt(n))) #t hitung

[1] -10.31843

> (t.lower = qt(alpha,df = n-1)) #t tabel eka arah

[1] -1.859548

>

> #p-value (bandingkan p-value dengan alpha)

> (pval.lower = pt(t, df=n-1)) #eka arah

[1] 3.356892e-06

>

> #Cara Otomatis

> t.test(x, mu=mu0, alternative = "less", conf.level = 0.95)

​

One Sample t-test

​

data: x

t = -10.318, df = 8, p-value = 3.357e-06

alternative hypothesis: true mean is less than 1.09

95 percent confidence interval:

-Inf 1.020774

sample estimates:

mean of x

1.005556

Console

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Solusi Soal 1

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Uji Hipotesis Selisih Rataan 2 Populasi

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Uji Hipotesis Selisih Rataan 2 Populasi

#Statistika uji (bandingkan t hitung dengan t tabel)

xbar = xbar1 – xbar2

z = (xbar-mu0)/sqrt((sigma1^2/n1)+(sigma2^2/n2)) #z hitung

z.lower = qnorm(alpha) #z tabel eka arah

z.upper = qnorm(1-alpha) #z tabel eka arah

z.half.alpha = qnorm(1-alpha/2) #z tabel dwi arah

z.twosided = c(-z.half.alpha, z.half.alpha)

​

#p-value (bandingkan p-value dengan alpha)

pval.lower = pnorm(z) #eka arah

pval.upper = pnorm(z, lower.tail = FALSE) #eka arah

pval.twosided = 2*pnorm(z) #dwi arah

#1. Kasus variansi 1 dan 2 diketahui

#Input

x1, x2 #data

xbar1 = mean(x1) #mean sampel x1

xbar2 = mean(x2) #mean sampel x2

mu0 #nilai hipotesis

sigma1, sigma2 #standar deviasi populasi

n1, n2 #banyak observasi

alpha = 0.05 #taraf signifikansi

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Uji Hipotesis Selisih Rataan 2 Populasi

#Cara Manual

df = n1+n2-2

Sp = (((n1-1_*S1^2)+((n2-1)*S2^2))/(df)

xbar = xbar1 – xbar2

t = (xbar-mu0)/(sqrt(Sp)*(sqrt((1/n1)+(1/n2)))) #t hitung

t.lower = qt(alpha, df) #t tabel eka arah

t.upper = qt(1-alpha, df) #t tabel eka arah

t.half.alpha = qt(1-alpha/2, df) #t tabel dwi arah

t.twosided = c(-t.half.alpha, t.half.alpha)

​

#p-value (bandingkan p-value dengan alpha)

pval.lower = pt(t, df) #eka arah

pval.upper = pt(t, df, lower.tail = FALSE) #eka arah

pval.twosided = 2*pt(t, df) #dwi arah

​

#Cara Otomatis

t.test(x1, x2, mu=mu0, var.equal=TRUE, alternative = c(“two.sided”,l”less”,”greater”), conf.level=0.95)

#2a. Kasus variansi 1 dan 2 tidak diketahui dan dianggap sama

#Input

x1, x2 #data

xbar1 = mean(x1) #mean sampel x1

xbar2 = mean(x2) #mean sampel x2

mu0 #nilai hipotesis

S1 = sd(x1) #standar deviasi sampel x1

S2 = sd(x2) #standar deviasi sampel x2

n1, n2 #banyak observasi

alpha = 0.05 #taraf signifikansi

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Uji Hipotesis Selisih Rataan 2 Populasi

#Cara Manual

df = ((S1^2/n1) + (S2^2/n2))^2/

(((1/(n1-1))*(S1^2/n1)^2)+((1/(n2-1))*(S2^2/n2)^2))

xbar = xbar1 - xbar2

t = (xbar-mu0)/(sqrt((S1^2/n1)+(S2^2/n2))) #t hitung

t.lower = qt(alpha, df) #t tabel eka arah

t.upper = qt(1-alpha, df) #t tabel eka arah

t.half.alpha = qt(1-alpha/2, df) #t tabel dwi arah

t.twosided = c(-t.half.alpha, t.half.alpha)

​

#p-value (bandingkan p-value dengan alpha)

pval.lower = pt(t, df=n-1) #eka arah

pval.upper = pt(t, df=n-1, lower.tail = FALSE) #eka arah

pval.twosided = 2*pt(t, df=n-1) #dwi arah

​

#Cara Otomatis

t.test(x1, x2, mu=mu0, var.equal=FALSE, alternative = c(“two.sided”,”less”,”greater”), conf.level=0.95)

#2b. Kasus variansi 1 dan 2 tidak diketahui dan dianggap berbeda

#Input

x1, x2 #data

xbar1 = mean(x1) #mean sampel x1

xbar2 = mean(x2) #mean sampel x2

mu0 #nilai hipotesis

S1 = sd(x1) #standar deviasi sampel x1

S2 = sd(x2) #standar deviasi sampel x2

n1, n2 #banyak observasi

alpha = 0.05 #taraf signifikansi

Click to add text

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Uji Hipotesis Rataan Berpasangan

#Cara Manual

t = (dbar – mu0)/(sd/sqrt(n)) #t hitung

​

t.lower = qt(alpha,df = n-1) #t tabel eka arah

t.upper = qt(1-alpha) #t tabel eka arah

t.half.alpha = qt(1-alpha/2) #t tabel dwi arah

t.twosided = c(-t.half.alpha, t.half.alpha)

​

#p-value (bandingkan p-value dengan alpha)

pval.lower = pt(t, df=n-1) #eka arah

pval.upper = pt(t, df=n-1, lower.tail = FALSE) #eka arah

pval.twosided = 2*pt(t, df=n-1) #dwi arah

​

#Cara Otomatis

t.test(x1, x2, mu=mu0, paired=T, alternative = c(“two.sided”,l”less”,”greater”), conf.level=0.95)

#3. Kasus Data Berpasangan

#Input

d = x1 - x2 #data

dbar = mean(d) #mean sampel

mu0 #nilai hipotesis

Sd = sd(d) #standar deviasi sampel

n = length(d) #banyak observasi

alpha = 0.05 #taraf signifikansi

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Contoh Soal 2

Sebuah majalah tentang kriminalitas menyatakan bahwa lama ditahannya (dalam bulan) seorang narapidana karena kasus penipuan lebih pendek setidaknya 10 bulan dibanding kasus senjata api. Seorang ahli kriminologi mencatat masa tahanan 10 narapidana karena kasus penipuan dan 8 narapidana karena kasus senjata api sampai mereka bebas dari penjara sbb:

​

​

​

​

Asumsikan bahwa data berasal dari distribusi normal. Untuk tingkat signifikansi 5%, apakah pernyataan pada majalah tersebut didukung oleh data yang ada?

Copyright 2020 © KK Statistika, FMIPA – ITB

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Prosedur R Soal 2

setwd("D://Praktikum Statistika//Praktikum 2020")

library(readxl)

#Input

x = read_excel("DATA UJI HIPOTESIS.xlsx", sheet = "latihan no 6")

x1 = as.numeric(x$Penipuan)

x2 = as.numeric(x$`Senjata Api`)

x2 = na.omit(x2) #menghapus missing value atau NA

xbar1 = mean(x1) #mean sampel x1

xbar2 = mean(x2) #mean sampel x2

mu0 = -10 #nilai hipotesis

S1 = sd(x1) #standar deviasi sampel x1

S2 = sd(x2) #standar deviasi sampel x2

n1 = length(x1) #banyak observasi

n2 = length(x2)

alpha = 0.05 #taraf signifikansi

​

#Cara Manual (Bandingkan Statistik Uji T hitung dan T tabel)

(df = ((S1^2/n1) + (S2^2/n2))^2/

(((1/(n1-1))*(S1^2/n1)^2)+((1/(n2-1))*(S2^2/n2)^2)))

(xbar = xbar1 - xbar2)

(t = (xbar-mu0)/(sqrt((S1^2/n1)+(S2^2/n2)))) #t hitung

(t.lower = qt(alpha, df)) #t tabel eka arah

​

#p-value (bandingkan p-value dengan alpha)

(pval.lower = pt(t, df)) #eka arah

Editor

> #Cara Manual (Bandingkan Statistik Uji T hitung dan T tabel)

> (df = ((S1^2/n1) + (S2^2/n2))^2/

+ (((1/(n1-1))*(S1^2/n1)^2)+((1/(n2-1))*(S2^2/n2)^2)))

[1] 12.60011

> (xbar = xbar1 - xbar2)

[1] -8.845

> (t = (xbar-mu0)/(sqrt((S1^2/n1)+(S2^2/n2)))) #t hitung

[1] 0.5318017

> (t.lower = qt(alpha, df)) #t tabel eka arah

[1] -1.775241

>

> #p-value (bandingkan p-value dengan alpha)

> (pval.lower = pt(t, df)) #eka arah

[1] 0.6979432

Console

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Prosedur R Soal 2

#Cara Otomatis

t.test(x1, x2, mu=mu0, var.equal=FALSE, alternative = "less", conf.level=0.95)

Editor

> #Cara Otomatis

> t.test(x1, x2, mu=mu0, var.equal=FALSE, alternative = "less", conf.level=0.95)

​

Welch Two Sample t-test

​

data: x1 and x2

t = 0.5318, df = 12.6, p-value = 0.6979

alternative hypothesis: true difference in means is less than -10

95 percent confidence interval:

-Inf -4.98942

sample estimates:

mean of x mean of y

8.830 17.675

Console

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Solusi Soal 2

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Uji Hipotesis Variansi 1 Populasi

#Cara Manual (Bandingkan chi hitung dan chi tabel)

chi = (n-1)*S^2/sigma0 #chi hitung

chi.lower = qchisq(alpha, df=n-1) #chi tabel eka arah

chi.upper = qchisq(1-alpha, df=n-1) #chi tabel eka arah

chi.half.alpha = qchisq(1-alpha/2, df=n-1) #chi tabel dwi arah

chi.twosided = c(-chi.half.alpha, chi.half.alpha)

​

#P-value (bandingkan p-value dengan alpha)

pval.lower = pchisq(chi, df=n-1) #eka arah

pval.upper = pchisq(chi, df=n-1, lower.tail = FALSE) #eka arah

pval.twosided = 2*pchisq(chi, df=n-1) #dwi arah

​

#Cara Otomatis

library(TeachingDemos)

sigma.test(x, sigma=sqrt(sigma0), alternative = c(“two.sided”, “less”, “greater”), conf.level = 0.95)

#Variansi satu populasi

#Input

x #data

sigma0 #nilai hipotesis

S = sd(x) #standar deviasi sampel x

n = length(d) #banyak observasi

alpha = 0.05 #taraf signifikansi

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Uji Hipotesis Variansi 2 Populasi

#Cara Manual (Bandingkan F hitung dan F tabel)

F = S1^2/S2^2 #F hitung

F.lower = qf(alpha, n1-1, n2-1) #chi tabel eka arah

F.upper = qf(1-alpha, n1-1, n2-1) #chi tabel eka arah

F.half.alpha = qf(1-alpha/2, n1-1, n2-1) #chi tabel dwi arah

F.twosided = c(-F.half.alpha, F.half.alpha)

​

#P-value (bandingkan p-value dengan alpha)

pval.lower = pf(F, n1-1, n2-1) #eka arah

pval.upper = pf(F, n1-1, n2-1, lower.tail = FALSE) #eka arah

pval.twosided = 2*pf(F, n1-1, n2-1) #dwi arah

​

#Cara Otomatis

var.test(x1, x2, ratio = 1, alternative = c(“two.sided”, “less”, “greater”), conf.level = 0.95)

#Variansi satu populasi

#Input

x1, x2 #data

S1 = sd(x1) #standar deviasi sampel x1

S2 = sd(x2) #standar deviasi sampel x2

n1 = length(x1) #banyak observasi x1

n2 = length(x2) #banyak observasi x2

alpha = 0.05 #taraf signifikansi

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Contoh Soal 3

Copyright 2020 © KK Statistika, FMIPA – ITB

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Prosedur R Soal 2

> #Cara Manual (Bandingkan chi hitung dan chi tabel)

> (chi = (n-1)*S^2/sigma0)

[1] 48.22222

> (chi.upper = qchisq(1-alpha, df=n-1)) #chi tabel eka arah

[1] 15.50731

> #P-value (bandingkan p-value dengan alpha)

> (pval.upper = pchisq(chi, df=n-1, lower.tail = FALSE)) #eka arah

[1] 8.958596e-08

>

> #Cara Otomatis

> library(TeachingDemos)

> sigma.test(x, sigma=sqrt(sigma0), alternative = "greater", conf.level = 0.95)

​

One sample Chi-squared test for variance

​

data: x

X-squared = 48.222, df = 8, p-value = 8.959e-08

alternative hypothesis: true variance is greater than 1e-04

95 percent confidence interval:

0.0003109644 Inf

sample estimates:

var of x

0.0006027778

library(readxl)

x <-read_excel("DATA UJI HIPOTESIS.xlsx", sheet = "contoh diameter logam")

x <- as.numeric(x$`diameter potongan logam`)

S = sd(x) #stadar deviasi sampel

sigma0 = 0.01^2 #nilai hipotesis variansi

n = length(x) #banyak data

alpha=0.05

#Cara Manual (Bandingkan chi hitung dan chi tabel)

(chi = (n-1)*S^2/sigma0) #chi hitung

(chi.upper = qchisq(1-alpha, df=n-1)) #chi tabel eka arah

​

#P-value (bandingkan p-value dengan alpha)

(pval.upper = pchisq(chi, df=n-1, lower.tail = FALSE)) #eka arah

​

#Cara Otomatis

library(TeachingDemos)

sigma.test(x, sigma=sqrt(sigma0), alternative = "greater", conf.level = 0.95)

Editor

Console

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Solusi Soal 3

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Uji Hipotesis Lainnya

UJI KENORMALAN

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UJI KENORMALAN

x<-c(5.6,5.4,4.3,7.4,5,6.67,6.3,4.8,

7.62,4.56,6.43,5.5) #input data

ks.test(x,"pnorm") #pnorm=distribusi normal

Contoh: produksi sumur minyak tahun 1992 di suatu daerah tercatat sebagai berikut:

​

​

​

​

Lakukan uji kenormalan terhadap data di atas dengan uji Kolmogorov-Smirnov.

> ks.test(x,"pnorm") #pnorm=distribusi normal

​

One-sample Kolmogorov-Smirnov test

​

data: x

D = 0.99999, p-value = 3.331e-16

alternative hypothesis: two-sided

Editor

Console

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Uji Hipotesis Lainnya

UJI KEBEBASAN

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UJI KEBEBASAN

> x<-c(27,13,35,15,33,27,25,25)

> (s<-matrix(x,3,4))

[,1] [,2] [,3] [,4]

[1,] 27 35 33 25

[2,] 13 15 27 25

> chisq.test(s)

​

Pearson's Chi-squared test

​

data: s

X-squared = 5.7292, df = 3, p-value = 0.1256

Seorang peneliti ingin mengethaui apakah terdapat hubungan antara jenis kelamin dengan hobi dengan data berikut:

INPUT DATA DAN BUAT TABEL KONTINGENSI

x<-c(27,13,35,15,33,27,25,25)

(s<-matrix(x,3,4)) #buat tabel kontingensi

chisq.test(s) #uji kebebasan

Console

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Tim Penyusun

Pengajar Semester I – 2020/2021

Dr. Udjianna S. Pasaribu

Dosen KK Statistika, MA2181 Analisis Data

Dr. Utriweni Mukhaiyar

Dosen KK Statistika, MA2082 Biostatistika

Dr. Sandy Vantika

Dosen KK Statistika,

MA2181 Analisis Data / MA2081 Statistika Dasar

Dr. Rr. Kurnia Novita Sari

Dosen KK Statistika, MA2181 Analisis Data

Dr. Sapto Wahyu Indratno

Dosen KK Statistika, MA2082 Biostatistika

Yuli Sri Afrianti, S.Si., MT, MBA.

Dosen KK Statistika,

MA2181 Analisis Data / MA2081 Statistika Dasar

Copyright 2020 © KK Statistika, FMIPA – ITB

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Selamat Praktikum!

Copyright 2020 © KK Statistika, FMIPA – ITB

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