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Matakuliah
Tahun
Versi
: I0014 / Biostatistika
: 2005
: V1 / R1
Pertemuan 11
Pengujian Hipotesis (I)
1
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)
2
Outline Materi
• Pendugaan Nilai tengah
( )
• Pendugaan beda dua nilai tengah
(1  2 )
3
<<ISI>>
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.
4
<<ISI>>
Logika Pengujian Hipotesis
A contingency table illustrates the possible outcomes
of a statistical hypothesis test.
5
<<ISI>>
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)
6
<<ISI>>
Pengujian Mean Populasi (n besar)
Null Hypothesis
H0:  = 0
Alternative Hypothesis H0:   0
Significance Level of
the Test
often 0.05 or 0.01)
Test Statistic
z
Critical Points
Critical Points of z

z

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
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
7
<<ISI>>
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
8
<<ISI>>
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
9
<<ISI>>
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 unknown10
<<ISI>>
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
11
<<ISI>>
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).
12
<<ISI>>
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
• has approximately a t
distribution with degrees of
freedom given by (round
downward to the nearest integer
if necessary):
t
( x1  x2 )  ( 1   2 )0
s12 s22

n1 n2
2
2
s22 
 s1
 n1  n2 

df  2 2
2
2
 s1 
 s2 
 
 
 n1 
n 
 2
n1  1 n2  1
13
<<ISI>>
Menggunakan Ragam
gabungan (Pooled Variance)
The estimate of the standard deviation of (x1  x 2 ) is given by:
1 
2 1
sp 


 n1 n 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.
14
<< CLOSING>>
• 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
15
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