Matakuliah
: I0014 / Biostatistika
Tahun
: 2008
Pengujian Hipotesis (I)
Pertemuan 11
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa dapat menjelaskan konsep pengujian
hipotesis (C2)
• Mahasiswa dapat menguji hipotesis untuk nilai
tengah (C3)
Bina Nusantara
Outline Materi
• Pendugaan Nilai tengah
( )
( 1  2 )
• Pendugaan beda dua nilai tengah
Bina Nusantara
Pengujian Hipotesis
•
A null hypothesis, denoted by H0, is an assertion about one or
more population parameters. This is the assertion we hold to be
true until we have sufficient statistical evidence to conclude
otherwise.
– H0: =100
• The alternative hypothesis, denoted by H1, is the assertion of all
situations not covered by the null hypothesis.
– H1: 100
• H0 and
H1 are:
– Mutually exclusive
– Only one can be true.
– Exhaustive
– Together they cover all possibilities, so one or the other
must be true.
Bina Nusantara
Logika Pengujian Hipotesis
A contingency table illustrates the possible outcomes of a
statistical hypothesis test.
Bina Nusantara
Kesalahan dalam Uji Hipotesis
• A decision may be incorrect in two ways:
– Type I Error: Reject a true H0
• The Probability of a Type I error is denoted by .
•  is called the level of significance of the test
– Type II Error: Accept a false H0
• The Probability of a Type II error is denoted by .
• 1 -  is called the power of the test.
•  and  are conditional probabilities:
 = P(Reject H 0 H 0 is true)
 = P(Accept H 0 H 0 is false)
Bina Nusantara
Pengujian Mean Populasi (n
besar)
Null Hypothesis
H0:  = 0
Alternative Hypothesis H0:   0
Critical Points of z

z

Significance Level of
the Test
often 0.05 or 0.01)
Test Statistic
z
Critical Points
2
x  0
(assuming  is unknown,
s
otherwise substitute  for s)
n
The bounds z that capture an area of (1-)
2
Decision Rule
Reject the null hypothesis if
either z > z a or z < -z a
2
Bina Nusantara
2
0.01
0.02
0.05
0.10
0.20
0.005
0.010
0.025
0.050
0.100
2
2.576
2.326
1.960
1.645
1.282
Pengujian Mean Populasi (n
kecil)
When the population is normal, the population standard deviation,, is unknown
and the sample size is small, the hypothesis test is based on the t distribution, with
(n-1) degrees of freedom, rather than the standard normal distribution.
Small - sample test statistic for the population mean, :
x - 0
t=
s
n
When the population is normally distributed and the null
hypothesis is true, the test statistic has a t distribution with
n -1 degrees of freedom
Bina Nusantara
Uji mean berpasangan (pair t
test)
Test statistic for the paired - observations t test:
D  D
t
sD
n
where D is the sample average difference between each
pair of observations, sD is the sample standard deviation
of these differences, and the sample size, n, is the number
of pairs of observations. The symbol  D is the population
mean difference under the null hypothesis. When the null
hypothesis is true and the population mean difference is  D ,
the statistic has a t distribution with (n - 1) degrees of freedom.
0
0
0
Bina Nusantara
Uji Mean Dua Populasi
Independen
When paired data cannot be obtained, use independent
random samples drawn at different times or under different
circumstances.
– Large sample test if:
• Both n1  30 and n2  30 (Central Limit Theorem), or
• Both populations are normal and 1 and 2 are both
known
– Small sample test if:
• Both populations are normal and 1 and 2 are
Bina Nusantara
•
Situasi Pengujian Dua Mean
Populasi
I: Difference between two population means is 0
• H0: 1 -2 = 0
• H1: 1 -2  0
•
II: Difference between two population means is less than 0
• H0: 1 -2  0
• H1: 1 -2  0
•
III: Difference between two population means is less than D
• H0: 1 -2  D
• H1: 1 -2  D
Bina Nusantara
Statistik Uji Dua Mean Populasi
Large-sample test statistic for the difference between two
population means:
( x1  x2 )  (  1   2 ) 0
z
s12 s22

n1 n2
The term (1- 2)0 is the difference between 1 an 2 under the
null hypothesis. Is is equal to zero in situations I and II, and it is
equal to the prespecified value D in situation III. The term in the
denominator is the standard deviation of the difference between
the two sample means (it relies on the assumption that the two
samples are independent).
Bina Nusantara
Uji Dua Mean Populasi dengan
Ukuran Contoh Kecil
• When sample sizes are small (n1<
30 or n2< 30 or both), and both
populations are normally
distributed, the test statistic
t
( x1  x2 )  ( 1   2 ) 0
s12 s22

n1 n2
2
• has approximately a t distribution
with degrees of freedom given by
(round downward to the nearest
integer if necessary):
Bina Nusantara
2
s22 
 s1
 n1  n2 

df  2 2
2
2
 s1 
 s2 
 
 
 n1 
 n2 

n1  1 n2  1
Menggunakan Ragam gabungan
(Pooled Variance)
The estimate of the standard deviation of (x1  x 2 ) is given by:
1 
2 1
sp 


n
n
 1 2
Test statistic for the difference between two population means, assuming equal
population variances:
(x1  x 2 )  (  1   2 ) 0
t=
1 
2 1
sp 


 n1 n2 
where (  1   2 ) 0 is the difference between the two population means under the null
hypothesis (zero or some other number D).
The number of degrees of freedom of the test statistic is df = ( n1  n2  2 ) (the
2
number of degrees of freedom associated with s p , the pooled estimate of the
population variance.
Bina Nusantara
Penutup
• Sampai saat ini Anda telah mempelajari
pengujian hipotesis nilai tengah, baik untuk
satu populasi maupun dua populasi
• Untuk dapat lebih memahami penggunaan
pengujian hipotesis tersebut, cobalah Anda
pelajari materi penunjang, dan mengerjakan
latihan
Bina Nusantara