Pengujian Hipotesis Dua Populasi By. Nurvita Arumsari, Ssi, MSi.
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Transcript of Pengujian Hipotesis Dua Populasi By. Nurvita Arumsari, Ssi, MSi.
![Page 1: Pengujian Hipotesis Dua Populasi By. Nurvita Arumsari, Ssi, MSi.](https://reader036.fdocuments.net/reader036/viewer/2022062322/56649ea95503460f94bad5ca/html5/thumbnails/1.jpg)
Statistik TP608377 A
Pengujian Hipotesis Dua PopulasiBy. Nurvita Arumsari, Ssi, MSi
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Chapter Goals
After completing this chapter, you should be able to:
Test hypotheses or form interval estimates for
◦two independent population means Standard deviations known Standard deviations unknown
◦two means from paired samples◦the difference between two population
proportions
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Estimating two population values
Population means, independent samples
Paired samples
Population proportions
Group 1 vs. independent Group 2
Same group before vs. after treatment
Proportion 1 vs. Proportion 2
Examples:
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Two – Sample Mean Population Test
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Hypothesis Tests forTwo Population Means
Lower tail test:
H0: μ1 = μ2
HA: μ1 < μ2
i.e.,
H0: μ1 – μ2 = 0HA: μ1 – μ2 < 0
Upper tail test:
H0: μ1 = μ2
HA: μ1 > μ2
i.e.,
H0: μ1 – μ2 = 0HA: μ1 – μ2 > 0
Two-tailed test:
H0: μ1 = μ2
HA: μ1 ≠ μ2
i.e.,
H0: μ1 – μ2 = 0HA: μ1 – μ2 ≠ 0
Two Population Means, Independent Samples
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Two Population Means, Independent Samples
Lower tail test:
H0: μ1 – μ2 = 0HA: μ1 – μ2 < 0
Upper tail test:
H0: μ1 – μ2 = 0HA: μ1 – μ2 > 0
Two-tailed test:
H0: μ1 – μ2 = 0HA: μ1 – μ2 ≠ 0
a a/2 a/2a
-za -za/2za za/2
Reject H0 if z < -za
Reject H0 if z > za
Reject H0 if z < -za/2
or z > za/2
Hypothesis tests for μ1 – μ2
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Hypothesis tests for μ1 – μ2
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
Use a z test statistic
Use s to estimate unknown σ , approximate with a z test statistic
Use s to estimate unknown σ , use a t test statistic and pooled standard deviation
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Independent Samples
Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
Different data sources◦ Unrelated◦ Independent Sample selected from
one population has no effect on the sample selected from the other population
Use the difference between 2 sample means
Use z test or pooled variance t test
*
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σ1 and σ2 known
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 known
Assumptions:
Samples are randomly and independently drawn
population distributions are normal or both sample sizes are 30
Population standard deviations are known
*
When σ1 and σ2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a z-value…
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
2
22
1
21
2121
nσ
nσ
μμxxz
The test statistic for μ1 – μ2 is:
σ1 and σ2 known
*
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
2
22
1
21
/221n
σ
n
σzxx
The confidence interval for μ1 – μ2 is:
σ1 and σ2 known(continued)
*
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σ1 and σ2 unknown, large samples
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, large samples
Assumptions:
Samples are randomly and independently drawn
both sample sizes are 30
Population standard deviations are unknown*
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, large samples
Forming interval estimates:
use sample standard deviation s to estimate σ
the test statistic is a z value
(continued)
*
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
2
22
1
21
/221n
s
n
szxx
The confidence interval for μ1 – μ2 is:
σ1 and σ2 unknown, large samples (continued)
*
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σ1 and σ2 unknown, small samples
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, small samples
Assumptions:
populations are normally distributed
the populations have equal variances
samples are independent
*
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, small samples
Forming interval estimates:
The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ
the test statistic is a t value with (n1 + n2 – 2) degrees of freedom
(continued)
*
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, small samples
The pooled standard deviation is
(continued)
2nn
s1ns1ns
21
222
211
p
*
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
21
p/221n
1
n
1stxx
The confidence interval for μ1 – μ2 is:
σ1 and σ2 unknown, small samples (continued)
Where t/2 has (n1 + n2 – 2) d.f.,
and
2nn
s1ns1ns
21
222
211
p
*
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Population means, independent samples
σ1 and σ2 known
σ1 and σ2 unknown, n1 and n2 30
σ1 and σ2 unknown, n1 or n2 < 30
σ1 and σ2 unknown, small samples
Where t/2 has (n1 + n2 – 2) d.f.,
and
2nn
s1ns1ns
21
222
211
p
21p
2121
n1
n1
s
μμxxt
The test statistic for μ1 – μ2 is:
*
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Paired Samples
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Lower tail test:
H0: μd = 0HA: μd < 0
Upper tail test:
H0: μd = 0HA: μd > 0
Two-tailed test:
H0: μd = 0HA: μd ≠ 0
Paired Samples
Hypothesis Testing for Paired Samples
a a/2 a/2a
-ta -ta/2ta ta/2
Reject H0 if t < -ta
Reject H0 if t > ta Reject H0 if t < -t /2a
or t > t /2a Where t has n - 1 d.f.
(continued)
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Paired Samples
Tests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values:
Eliminates Variation Among Subjects Assumptions:
◦ Both Populations Are Normally Distributed◦ Or, if Not Normal, use large samples
d = x1 - x2
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Paired Differences
The ith paired difference is di , wherePaired samples di = x1i - x2i
The point estimate for the population mean paired difference is d :
1n
)d(ds
n
1i
2i
d
n
dd
n
1ii
The sample standard deviation is
n is the number of pairs in the paired sample
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Paired Differences
The confidence interval for d isPaired samples
1n
)d(ds
n
1i
2i
d
n
std d/2
Where t/2 has
n - 1 d.f. and sd
(continued)
n is the number of pairs in the paired sample
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Two Population Proportions
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Goal: Form a confidence interval for or test a hypothesis about the difference between two population proportions, p1 – p2
The point estimate for the difference is p1 – p2
Population proportions
Assumptions: n1p1 5 , n1(1-p1) 5
n2p2 5 , n2(1-p2) 5
Two Population Proportions
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Hypothesis Tests forTwo Population Proportions
Population proportions
Lower tail test:
H0: p1 – p2 = 0HA: p1 – p2 < 0
Upper tail test:
H0: p1 – p2 = 0HA: p1 – p2 > 0
Two-tailed test:
H0: p1 – p2 = 0HA: p1 – p2 ≠ 0
a a/2 a/2a
-za -za/2za za/2
Reject H0 if z < -za Reject H0 if z > za Reject H0 if z < -z /2a
or z > z /2a
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Two Population Proportions
Population proportions
21
21
21
2211
nn
xx
nn
pnpnp
The pooled estimate for the overall proportion is:
where x1 and x2 are the numbers from samples 1 and 2 with the characteristic of interest
Since we begin by assuming the null hypothesis is true, we assume p1 = p2 and pool the two p
estimates
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Two Population Proportions
Population proportions
21
2121
n1
n1
)p1(p
ppppz
The test statistic for p1 – p2 is:
(continued)
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Confidence Interval forTwo Population
Proportions
Population proportions
2
22
1
11/221 n
)p(1p
n
)p(1pzpp
The confidence interval for p1 – p2 is:
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Contoh Kasus
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Pooled sp t Test: Example
You’re a welding inspector. Is there a difference in level of precision between welding machine A & B? You collect the following data: A BNumber 21 25Sample mean 3.27 2.53Sample std dev 1.30 1.16
Assuming equal variances, isthere a difference in average yield ( = 0.05)?
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Calculating the Test Statistic
1.2256
22521
1.161251.30121
2nn
s1ns1ns
22
21
222
211
p
2.040
251
211
1.2256
02.533.27
n1
n1
s
μμxxt
21p
2121
The test statistic is:
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Solution
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
HA: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
= 0.05df = 21 + 25 - 2 = 44Critical Values: t = ± 2.0154
Test Statistic:
Decision:
Conclusion:
Reject H0 at a = 0.05There is evidence of a difference in means.
t0 2.0154-2.0154
.025
Reject H0 Reject H0
.025
2.040
040.2
251
211
2256.1
53.227.3
t
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Assume you are a welding inspector. In the first inspection, you found some defects in welding result. Then, you got a welding training. After you got a training, you inspect the welding result, like in table. Is the training effective? You collect the following data:
Paired Samples Example
Welding ins.
Number of Defects Difference
Before After
C.B 20 3 -17
T.F 20 6 -14
M.H 15 4 -11
R.K 10 4 -6
M.O 17 7 -10
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Assume you are a welding inspector. In the first inspection, you found some defects in welding result. Then, you got a welding training. After you got a training, you inspect the welding result, like in table. Is the training effective? You collect the following data:
d = di
n
159.41n
)d(ds
2i
d
= -11.2
Welding ins.
Number of Defects Difference
Before After
C.B 20 3 -17
T.F 20 6 -14
M.H 15 4 -11
R.K 10 4 -6
M.O 17 7 -10
Paired Samples Example
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Example: Two population Proportions
Is there a significant difference between the proportion of boy and the proportion of girl who will prefer on Welding department?
In a random sample, 52 of 72 boys and 10 of 50 girls indicated they would prefer welding department
Test at the .05 level of significance
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The hypothesis test is:
H0: p1 – p2 = 0 (the two proportions are equal)HA: p1 – p2 ≠ 0 (there is a significant difference between proportions)
The sample proportions are:
Boys: p1 = 52/72 = .72
Girls: p2 = 10/50 = .2
.508122
62
5072
0125
nn
xxp
21
21
The pooled estimate for the overall proportion is:
Example: Two population
Proportions(continued)