050222190613kuliah statistik 9 - tes hipotesis dua populasi
Post on 04-Jun-2018
221 Views
Preview:
TRANSCRIPT
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
1/37
JURUS N TEKNIK SIPIL
UNIVERSIT S ND L S
oleh :Purnawan, PhD ----- Kuliah ke 9 -----
ST TISTIKdan
PROB BILIT S
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
2/37
Chap 9-2Chap 9-2
Chapter 9 Estimation and Hypothesis Testing
for Two Population Parameters
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
3/37
Chap 9-3
Chapter Goals
After completing this chapter, you should beable to:
Test hypotheses or form interval estimates fortwo independent population means
Standard deviations knownStandard deviations unknown
the difference between two populationproportions
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
4/37
Chap 9-4
Estimation for Two Populations
Estimating twopopulation values
Populationmeans,
independentsamples
Populationproportions
Group 1 vs.independentGroup 2
Proportion 1 vs.Proportion 2
Examples:
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
5/37
Chap 9-5
Difference Between Two Means
Population means,independent
samples
1 and 2 known
1 and 2 unknown,n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
Goal: Form a confidenceinterval for the difference
between two populationmeans, 1 2
The point estimate for the
difference is
x1 x2
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
6/37
Chap 9-6
Independent Samples
Population means,independent
samples
1 and 2 known
1 and 2 unknown,n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
Different data sourcesUnrelatedIndependent
Sample selected fromone population has noeffect on the sampleselected from the otherpopulation
Use the difference between 2sample meansUse z test or pooled variancet test
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
7/37Chap 9-7
Population means,independent
samples
1 and 2 known
1 and 2 unknown,n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
1 and 2 known
Assumptions:
Samples are randomly andindependently drawn
population distributions arenormal or both sample sizes
are 30
Population standarddeviations are known
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
8/37
Chap 9-8
Population means,independent
samples
1 and 2 known
1 and 2 unknown,n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
and the standard error ofx1 x2 is
When 1 and 2 are known andboth populations are normal orboth sample sizes are at least 30,
the test statistic is a z- value
2
22
1
21
xx n
n
21
(continued)
1 and 2 known
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
9/37
Chap 9-9
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
2
22
1
21
/221
n
n
zxx
The confidence interval for1 2 is:
1 and 2 known(continued)
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
10/37
Chap 9-10
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
1 and 2 unknown, large samples
Assumptions:
Samples are randomly andindependently drawn
both sample sizesare 30
Population standarddeviations are unknown*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
11/37
Chap 9-11
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
1 and 2 unknown, large samples
Forming intervalestimates:
use sample standarddeviation s to estimate
the test statistic is a z value
(continued)
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
12/37
Chap 9-12
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
2
22
1
21
/221
n
s
n
szxx
The confidence interval for1 2 is:
1 and 2 unknown, large samples(continued)
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
13/37
Chap 9-13
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
1 and 2 unknown, small samples
Assumptions:
populations are normally
distributed
the populations have equalvariances
samples are independent
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
14/37
Chap 9-14
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
1 and 2 unknown, small samples
Forming intervalestimates:
The population variancesare assumed equal, so usethe two sample standarddeviations and pool them to
estimate the test statistic is a t value
with (n 1 + n 2 2) degreesof freedom
(continued)
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
15/37
Chap 9-15
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
1 and 2 unknown, small samples
The pooled standarddeviation is
(continued)
2nn
s1ns1ns
21
222
211
p
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
16/37
Chap 9-16
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
21
p/221 n1
n1
stxx
The confidence interval for1 2 is:
1 and 2 unknown, small samples(continued)
Where t /2 has (n 1 + n 2 2) d.f.,
and
2nns1ns1n
s21
222
211
p*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
17/37
Chap 9-17
Hypothesis Tests for the DifferenceBetween Two Means
Testing Hypotheses about 1 2
Use the same situations discussed already:Standard deviations known or unknown
Sample sizes 30 or < 30
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
18/37
Chap 9-18
Hypothesis Tests forTwo Population Proportions
Lower tail test:
H0: 1 2 H A: 1 < 2
i.e.,
H0: 1 2 0H A: 1 2 < 0
Upper tail test:
H0: 1 2H A: 1 > 2
i.e.,
H0: 1 2 0H A: 1 2 > 0
Two-tailed test:
H0: 1 = 2H A: 1 2
i.e.,
H0: 1 2 = 0H A: 1 2 0
Two Population Means, Independent Samples
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
19/37
Chap 9-19
Hypothesis tests for 1 2
Population means, independent samples
1 and 2 known
1 and 2 unknown,n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
Use a z test statistic
Use s to estimate unknown , approximate with a z test statistic
Use s to estimate unknown , use a t test statistic andpooled standard deviation
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
20/37
Chap 9-20
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
2
2
2
1
2
1
2121
n
n
xxz
The test statistic for1 2 is:
1 and 2 known
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
21/37
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
22/37
Chap 9-22
Population means,independent
samples
1 and 2 known
1 and
2 unknown,
n1 and n 2 30
1 and 2 unknown,n1 or n 2 < 30
1 and 2 unknown, small samples
Where t /2 has (n 1 + n 2 2) d.f.,
and
2nn
s1ns1ns
21
222
211
p
21
p
2121
n1
n1
s
xxt
The test statistic for1 2 is:
*
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
23/37
Chap 9-23
Two Population Means, Independent Samples
Lower tail test:
H0: 1 2 0
H A: 1 2 < 0
Upper tail test:
H0: 1 2 0
H A: 1 2 > 0
Two-tailed test:
H0: 1 2 = 0
H A: 1 2 0
/2 /2
-z -z /2 z z /2 Reject H 0 if z < -z Reject H 0 if z > z Reject H 0 if z < -z /2
or z > z /2
Hypothesis tests for 1 2
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
24/37
Chap 9-24
Pooled s p t Test: Example
To select the best stabilisation material. Is there a differencein the result of analysis on soil cement stabilisation and limestabilisation? The result of laboratory analysis is:
Cement LimeNumber 21 25Sample mean 3.27 2.53Sample std dev 1.30 1.16
Assuming equal variances, isthere a difference in averageyield ( = 0.05 )?
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
25/37
Chap 9-25
Calculating the Test Statistic
1.225622521
1.161251.301212nn
s1ns1ns
22
21
222
211
p
2.040
251
2111.2256
02.533.27
n1
n1s
xxz
21p
2121
The test statistic is:
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
26/37
Chap 9-26
Solution
H0: 1 - 2 = 0 i.e. ( 1 = 2)HA: 1 - 2 0 i.e. ( 1 2)
= 0.05
df = 21 + 25 - 2 = 44Critical Values: t = 2.0154
Test Statistic: Decision:
Conclusion:Reject H 0 at = 0.05
There is evidence of adifference in means.
t0 2.0154-2.0154
.025
Reject H 0 Reject H 0
.025
2.040
040.2
251
211
2256.153.227.3z
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
27/37
Chap 9-27
Two Population Proportions
Goal: Form a confidence interval foror test a hypothesis about thedifference between two population
proportions, p 1 p 2
The point estimate forthe difference is p1 p 2
Populationproportions
Assumptions: n1p1 5 , n 1(1-p 1) 5
n2p2 5 , n 2(1-p 2) 5
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
28/37
Chap 9-28
Confidence Interval forTwo Population Proportions
Populationproportions
222
1
11/221 n
)p(1pn
)p(1pzpp
The confidence interval forp1 p 2 is:
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
29/37
Chap 9-29
Hypothesis Tests forTwo Population Proportions
Population proportions
Lower tail test:
H0: p 1 p 2 H A: p 1 < p 2
i.e.,
H0: p 1 p 2 0H A: p 1 p 2 < 0
Upper tail test:
H0: p 1 p 2H A: p 1 > p 2
i.e.,
H0: p 1 p 2 0H A: p 1 p 2 > 0
Two-tailed test:
H0: p 1 = p 2H A: p 1 p 2
i.e.,
H0: p 1 p 2 = 0H A: p 1 p 2 0
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
30/37
Chap 9-30
Two Population Proportions
Populationproportions
21
21
21
2211
nnxx
nnpnpn
p
The pooled estimate for theoverall proportion is:
where x 1 and x 2 are the numbers fromsamples 1 and 2 with the characteristic of interest
Since we begin by assuming the nullhypothesis is true, we assume p 1 = p 2 and pool the two p estimates
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
31/37
Chap 9-31
Two Population Proportions
Populationproportions
21
2121
n1
n1)p1(p
ppppz
The test statistic forp1 p 2 is:
(continued)
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
32/37
Chap 9-32
Hypothesis Tests forTwo Population Proportions
Population proportions
Lower tail test:
H0: p 1 p 2 0
H A: p 1 p 2 < 0
Upper tail test:
H0: p 1 p 2 0
H A: p 1 p 2 > 0
Two-tailed test:
H0: p 1 p 2 = 0
H A: p 1 p 2 0
/2 /2
-z -z /2 z z /2 Reject H 0 if z < -z Reject H 0 if z > z Reject H 0 if z < -z /2
or z > z /2
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
33/37
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
34/37
Chap 9-34
The hypothesis test is:
H0: p 1 p 2 = 0 (the two proportions are equal)H A: p 1 p 2 0 (there is a significant difference between proportions)
The sample proportions are:Men: p 1 = 36/72 = .50
Women: p 2 = 31/50 = .62
.54912267
50723136
nnxx
p21
21
The pooled estimate for the overall proportion is:
Example:Two population Proportions
(continued)
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
35/37
Chap 9-35
The test statistic for p1 p 2 is:
Example:Two population Proportions
(continued)
.025
-1.96 1.96
.025
-1.31
Decision: Do not reject H 0
Conclusion: There is notsignificant evidence of adifference in proportionswho will vote yes between
men and women.
1.31
50
1
72
1.549)(1.549
0.62.50
n1n1p)(1p
ppppz
21
2121
Reject H 0 Reject H 0
Critical Values = 1.96 For = .05
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
36/37
Chap 9-36
Two Sample Tests in EXCEL
For independent samples:
Independent sample Z test with variances known:Tools | data analysis | z-test: two sample for means
Independent sample Z test with large sampleTools | data analysis | z-test: two sample for means
If the population variances are unknown, use sample variances
For paired samples (t test): Tools | data analysis | t -test: paired two sample for means
-
8/13/2019 050222190613Kuliah Statistik 9 - Tes Hipotesis Dua Populasi
37/37
Chapter Summary
Compared two independent samplesFormed confidence intervals for the differences between twomeansPerformed Z test for the differences in two means
Performed t test for the differences in two means
Compared two population proportionsFormed confidence intervals for the difference between twopopulation proportions
Performed Z-test for two population proportions
top related