Matakuliah
Tahun
: I0272 - STATISTIK PROBABILITAS
: 2009
PENGUJIAN HIPOTESIS 2
Pertemuan 10
Materi
• Pengujian hipotesis proporsi
• Pengujian hipotesis ragam (varian)
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3
Proportion
• Involves categorical values
• Two possible outcomes
– “Success” (possesses a certain characteristic) and
“Failure” (does not possesses a certain
characteristic)
• Fraction or proportion of population in the “success”
category is denoted by p
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4
4
Proportion
(continued)
• Sample proportion in the success category is denoted by
pS
–
X Number of Successes
ps  
n
Sample Size
• When both np and n(1-p) are at least 5, pS can be
approximated by a normal distribution with mean and
standard deviation
–
p  p
s
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p(1  p)
 ps 
n
5
5
• Test Statistic:
Z
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pS  p
p 1  p 
n

.05  .04
.04 1  .04 
500
 1.14
66
Example: Z Test for Proportion
Q. A marketing company
claims that it receives 4%
responses from its
mailing. To test this
claim, a random sample of
500 were surveyed with
25 responses. Test at the
a = .05 significance level.
Check:
np  500 .04   20
5
n 1  p   500 1  .04 
 480  5
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7
Z Test for Proportion: Solution
H0: p  .04
H1: p  .04
Test Statistic:
Z
a = .05
n = 500
.025
-1.96
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p 1  p 
n

.05  .04
.04 1  .04 
500
 1.14
Decision:
Critical Values:  1.96
Reject
pS  p
Do not reject at a = .05
Reject
.025
0 1.96 Z
1.14
Conclusion:
We do not have sufficient
evidence to reject the
company’s claim of 4%
response rate.
8
p -Value Solution
(p Value = 0.2542)  (a = 0.05). Do Not Reject.
p Value = 2 x .1271
Reject
Reject
a = 0.05
0
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1.14
1.96
Z
Test Statistic 1.14 is in the Do Not Reject Region
9
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 p0 is the hypothesized value of the
population proportion).
H0: p > p0
Ha: p < p0
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H0: p < p0
Ha: p > p0
H0: p = p0
Ha: p ≠ p0
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Pengujian Beda Dua Proporsi
Hipotesis nol dan alternatif :
H0 : p1 = p2 = 0
• H1 : p1 – p2 < 0, p1 – p2 > 0, p1 – p2  0
• Test statistic untuk n dan n besar:
1
z
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 pˆ 1  pˆ 2 
p1q1
p2 q2

n1
n2
2

ˆ1  p
ˆ2
p
1
1 

pq

 n1 n2 
1111
Hypothesis Testing
About a Population Variance
•
Left-Tailed Test
•Hypotheses
H0 :  2   02
H a :  2   02
where  02 is the hypothesized value
for the population variance
•Test Statistic
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 
2
( n  1) s 2
 20
12
Hypothesis Testing
About a Population Variance

Left-Tailed Test (continued)
•Rejection Rule
Critical value approach:
p-Value approach:
Reject H0 if  2  (12 a )
Reject H0 if p-value < a
where  (12 a ) is based on a chi-square
distribution with n - 1 d.f.
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Hypothesis Testing
About a Population Variance

Right-Tailed Test
•Hypotheses
H 0 :  2   20
H a :  2   20
where  02 is the hypothesized value
for the population variance
•Test Statistic
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2 
( n  1) s 2
 20
14
Hypothesis Testing
About a Population Variance

Right-Tailed Test (continued)
•Rejection Rule
Critical value approach:
p-Value approach:
Reject H0 if  2  a2
Reject H0 if p-value < a
where a2 is based on a chi-square
distribution with n - 1 d.f.
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15
Hypothesis Testing
About a Population Variance

Two-Tailed Test
•Hypotheses
H 0 :  2   20
H a :  2   20
where  02 is the hypothesized value
for the population variance
•Test Statistic
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2 
( n  1) s 2
 20
16
Hypothesis Testing
About a Population Variance

Two-Tailed Test (continued)
•Rejection Rule
Critical value approach:
Reject H0 if  2  (12 a /2) or  2  a2 /2
p-Value approach:
Reject H0 if p-value < a
where (12 a /2) and a2 /2 are based on a
chi-square distribution with n - 1 d.f.
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17
Hypothesis Testing About the
Variances of Two Populations

One-Tailed Test
•Hypotheses
H0 :  12   22
H a :  12   22
Denote the population providing the
larger sample variance as population 1.
•Test Statistic
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2
s
F 1
s22
18
Hypothesis Testing About the
Variances of Two Populations

One-Tailed Test (continued)
•Rejection Rule
Critical value approach:
Reject H0 if F > Fa
where the value of Fa is based on an
F distribution with n1 - 1 (numerator)
and n2 - 1 (denominator) d.f.
p-Value approach:
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Reject H0 if p-value < a
19
Hypothesis Testing About the
Variances of Two Populations

Two-Tailed Test
•Hypotheses
H 0 :  12   22
Ha : 12   22
Denote the population providing the
larger sample variance as population 1.
•Test Statistic
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2
s
F 1
s22
20
Hypothesis Testing About the
Variances of Two Populations

Two-Tailed Test (continued)
•Rejection Rule
Critical value approach:
Reject H0 if F > Fa/2
where the value of Fa/2 is based on an
F distribution with n1 - 1 (numerator)
and n2 - 1 (denominator) d.f.
p-Value approach:
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Reject H0 if p-value < a
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