Matakuliah : L0104 / Statistika Psikologi
Tahun : 2008
Pengujian Hipotesis Proporsi dan Varians/Ragam
Pertemuan 16

Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu :
Mahasiswa akan dapat menyusun simpulan dari hasil uji hipotesis proporsi dan varians/ragam
3
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Outline Materi
• Uji hipotesis proporsi sampel besar
• Uji hipotesis beda proporsi sampel besar
• Uji hipotesis varians
• Uji hipotesis kesamaan dua varians
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4

Hypothesis Testing
• Developing Null and Alternative Hypotheses
• Type I and Type II Errors
• One-Tailed Tests About a Population Mean:
Large-Sample Case
• Two-Tailed Tests About a Population Mean:
Large-Sample Case
• Tests About a Population Mean:
Small-Sample Case continued
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Hypothesis Testing
• Tests About a Population Proportion
• Hypothesis Testing and Decision Making
• Calculating the Probability of Type II Errors
• Determining the Sample Size for a Hypothesis
Test about a Population Mean
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Developing Null and Alternative Hypotheses
• Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected.
• The null hypothesis , denoted by H
0
, is a tentative assumption about a population parameter.
• The alternative hypothesis, denoted by H a
, is the opposite of what is stated in the null hypothesis.
• Hypothesis testing is similar to a criminal trial. The hypotheses are:
H
0
: The defendant is innocent
H a
: The defendant is guilty
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Developing Null and Alternative
Hypotheses
• Testing Research Hypotheses
– The research hypothesis should be expressed as the alternative hypothesis.
– The conclusion that the research hypothesis is true comes from sample data that contradict the null hypothesis .
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A Summary of Forms for Null and
Alternative Hypotheses about a
Population Mean
• The equality part of the hypotheses always appears in the null hypothesis.
• In general, a hypothesis test about the value of a population mean μ must take one of the following three forms (where μ
0 is the hypothesized value of the population mean).
H
0
: μ > μ
0
H a
: μ < μ
0
H
0
: μ < μ
0
H a
: μ > μ
0
H
0
: μ = μ
0
H a
: μ μ
0

Type I and Type II Errors
•
• Since hypothesis tests are based on sample data, we must allow for the possibility of errors.
• A Type I error is rejecting H
0 when it is true.
• A Type II error is accepting H
0 when it is false.
• The person conducting the hypothesis test specifies the maximum allowable probability of making a
Type I error, denoted by α and called the level of significance.
• Generally, we cannot control for the probability of making a Type II error, denoted by β.
• Statistician avoids the risk of making a Type II
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0
H
0
”.
” and not “accept

Contoh Soal: Metro EMS
• Type I and Type II Errors
Conclusion
12 )
Population Condition
H
0
True
( μ < 12 )
H a
True
( μ >
Accept H
0
Correct
(Conclude μ < 12 ) Conclusion
Error
Type II
Correct
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Reject H
0
Type I
(Conclude μ > 12) Error
Conclusion




The p-value is the probability of obtaining a sample result that is at least as unlikely as what is observed.
The p-value can be used to make the decision in a hypothesis test by noting that:
if the p-value is less than the level of significance  , the value of the test statistic is in the rejection region.
if the p-value is greater than or equal to  , the value of the test statistic is not in the rejection region.
Reject H
0 if the p-value <  .
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





Determine the appropriate hypotheses.
Select the test statistic for deciding whether or not to reject the null hypothesis.
Specify the level of significance  for the test.
Use  to develop the rule for rejecting H
0
.
Collect the sample data and compute the value of the test statistic.
a) Compare the test statistic to the critical value(s) in the rejection rule, or b) Compute the p-value based on the test statistic and compare it to whether or not to reject H
0
.
  to determine
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
Hypotheses
H
0
:   

H a
:   
 or H
0
:   

H a
:   


Test Statistic  Known z

/

0 n z
 Unknown

0 s / n

Rejection Rule
Reject H
0 if z > z

Reject H
0 if z < -z


Contoh Soal: Metro EMS
• One-Tailed Test about a Population Mean: Large n
Let = P (Type I Error) = .05
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Sampling distribution of (assuming H true and 
0
= 12) is
Do Not Reject H
0
Reject H
0
 
12
1.645
 x c
(Critical value) x

Contoh Soal: Metro EMS
• One-Tailed Test about a Population Mean: Large n
Let n x s = 3.2 minutes
(The sample standard deviation s can be used to estimate the population standard deviation α .) z
 x

/
  n

.
. / 40

Since 2.47 > 1.645, we reject H
0
.
Conclusion : We are 95% confident that Metro EMS is not meeting the response goal of 12 minutes; appropriate action should be taken to improve service.
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Contoh Soal: Metro EMS
• Using the p -value to Test the Hypothesis
Recall that z x p -value =
.0068.
Since p -value < α , that is .0068 < .05, we reject H
0
.
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Do Not Reject H
0
0
Reject H
0
p-value  
1.645
2.47
z

Two-Tailed Tests about a
Population Mean:
Large-Sample Case ( n > 30)
• Hypotheses
• Test Statistic
• Rejection Rule
H
0
: μ = μ
0
H a
: μ μ
σ Known σ Unknown z

/

0 n z
 s / n
0
Reject H
0 if | z | > z

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Summary of Test Statistics to be Used in a
Hypothesis Test about a Population Mean
Yes No n > 30 ?
No s known ?
Yes
Use s to estimate s s known ?
Yes
Yes
No
Use s to estimate s
Popul. approx.
normal
?
No z

 x
/
  n z
 x
  s / n z

 x
/
  n t
 x
  s / n
Increase n to > 30
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A Summary of Forms for Null and
Alternative Hypotheses about a
Population Proportion
• The equality part of the hypotheses always appears in the null hypothesis.
• In general, a hypothesis test about the value of a population proportion p must take one of the following three forms (where p
0 is the hypothesized value of the population proportion).
H
0
: p > p
0
H a
: p < p
0
H
0
: p < p
0
H a
: p > p
0
H
0
H a
: p = p
0
: p
 p
0
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A Summary of Forms for Null and
Alternative Hypotheses about a
Differen of two Population
Proportion
• The equality part of the hypotheses always appears in the null hypothesis.
• In general, a hypothesis test about the value of a population proportion p must take one of the following three forms
H
0
: p
1
> p
2
H a
: p
1
< p
2
H
0
: p
1
< p
2
H a
: p
1
> p
2
H
0
: p
1
= p
2
H a
: p
1
≠ p
2
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Contoh Soal: NSC
• Two-Tailed Test about a Population Proportion:
Large n
– Hypothesis
H
0
: p

H a
: p .5
– Test Statistic
 p
 p
0
(1 n
 p
0
)


120
 .045644
z 


0 p

.045644
 1.278
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Hypothesis Testing
About a Population Variance

Right-Tailed Test
• Hypotheses
 
 
• Test Statistic
( ( n

1 ) ) s s
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• Rejection Rule
Reject H
0
 
( ( 1
 
) )
 2 if (where is based on a chi-square distribution with n - 1 d.f.) or
Reject H
0 if p-value < 

Hypothesis Testing
About a Population Variance

Two-Tailed Test
• Hypotheses  
 
• Test Statistic
( ( n

1 ) ) s s
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• Rejection Rule
 
( ( 1
 
/ ) or
 
/ /
Reject H
0
 

2
/2
(where are based on a chi-square distribution with n - 1 d.f.) or Reject H
0 if p-value < 

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