10. bus stat - chapter 10 - april 4th
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Copyright 2011 Pearson Education, Inc. publishing as Prentice Hall 10-1
Business Statistics:
A Decision-Making Approach8thEdition
Chapter 10
Estimation and Hypothesis Testing
for Two Population Parameters
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Estimation for Two Populations
Estimating two
population values
Population
means,
independent
samples
Paired
samples
Population
proportions
Group 1 vs.Group 2
Same groupbefore vs. after
Proportion 1 vs.Proportion 2
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Difference Between Two Means
Population means,
independent
samples
1and 2known
1and 2unknownbut assumed equal
1and 2unknown,not assumed equal
Goal: Form a confidence
interval for the difference
between two populationmeans, 12
The point estimate for the
difference is
x1x2
*
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1and 2known
Assumptions:
Samples are randomly and
independently drawn
population distributions are
normal or both sample sizes
are 30
Population standard
deviations are known
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When 1and 2are known and both populations are normal or
both sample sizes are at least 30, the test statistic is a z
valueand the standard error of x1x2 is
2
22
1
21
xx n
n
21
(continued)
1and 2known
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Confidence Interval:
1and 2known
The confidence interval for
12 is:
2
2
2
1
2
1/221
n
n
xx z
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General Steps
1. Define the population parameter of interest
and select independent samples from each
population2. Specify the confidence interval
3. Compute the point estimate
4. Determine the standard error
5. Determine the critical value
6. Develop the confidence interval estimate
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Example 1 with std dev known
RS Royal Jemursari mengadakan survey untuk
mengetahui berapa lama pasien
dikonsultasikan dengan dokter. Rata-rata untuk
100 pria adalah 35 menit dengan std dev 11menit dan 100 wanita 42 menit dengan std dev
16 menit.
Cari 95% Confidence Interval
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Example 2 with std dev known
Perbandingan on time attendance di kelas
bisnis statistik antara kelas reguler dan IBM
telah disurvey. Rata-rata 150 kelas reguler
adalah 45 menit dengan std dev 15 menit dan100 siswa IBM 30 menit dng std dev 10 menit.
Cari 90 Confidence Interval
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Example 3 Std Dev known
Cari 95% Confidence interval estimate utuk
kedua populasi yg disediakan.
Populasi 1
Mean1 : 345
Std Dev1 : 34
N1 : 50
Populasi 2
Mean1 : 320
Std Dev1 : 40
N1 : 80
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Population means,
independent
samples
1and 2known
1and 2unknown,large samples
Assumptions: Samples are randomly and
independently drawn
Populations follow the normal
distribution
The two standard deviationsare equal*
1and
2unknown
but assumed equal
1and 2unknown,not assumed equal
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1and 2unknown,large samples
(continued)
Forming interval estimates:
The population standard deviations are
assumed equal, so use the two sample standarddeviations and pool them to estimate (called
pooled standard deviation)
the test statistic is a t value with (n1+ n22)degrees of freedom
d k
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1and 2unknown,large samples
(continued)
The pooled standard deviation is:
2nn
s1ns1ns
21
2
22
2
11p
The confidence interval for 12 is:
21p/221
n
1
n
1stxx
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Contoh 1, std dev unknown
Kalbe mengadakan penelitian utk obat terbaru
berapa lama obat itu akan larut dalam darah.
Rata rata dari 6 orang yang berumur diatas
50th adalah 13.6 menit, S1: 3.1 menit dan 8orang yang berumur lebih dari 50th adalah 11.2
menit, S2: 5 menit.
Cari 95% Confidence Interval
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Contoh 2, std dev unknown
Cari 90% Confidence Interval dari
Sample 1
N1: 15
X1: 50
S1: 5
Sample 2
N1: 13 X1: 53
S1: 6
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Population means,
independent
samples
1and 2known
1and 2unknown,small samples
Assumptions:
populations are normally
distributed
there is a reason to believe
that the populations do not
have equal variances
samples are independent
*
1and
2unknown
but assumed equal
1and 2unknown,not assumed equal
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1and 2unknown,small samples
Forming interval estimates:
The population variances
are not assumed equal, sowe do not pool them
the test statistic is a t value
with degrees of freedom
given by:
(continued)
1n
/ns
1n
/ns
)/ns/n(sdf
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1
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2
2
2
1
2
1/221
n
s
n
stxx
The confidence interval for 12 is:
1and 2unknown,small samples
(continued)
Where t/2 has d.f. given by
1n
/ns
1n
/ns
)/ns/n(sdf
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1
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Example 1
Visa
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Hypothesis Tests for theDifference Between Two Means
Testing Hypotheses about 12
Use the same situations discussed already: Standard deviations known
Standard deviations unknown
Assumed equal
Assumed not equal (small samples)
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Hypothesis Tests forTwo Population Means
Lower tail test:
H0: 12HA: 1< 2
i.e.,
H0: 120HA: 12< 0
Upper tail test:
H0: 12HA: 1> 2
i.e.,
H0: 120HA: 12> 0
Two-tailed test:
H0: 1= 2HA: 12
i.e.,
H0: 12= 0HA: 120
Two Population Means, Independent Samples
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Hypothesis tests for 12
Population means, independent samples
1and 2knownUse a z test statistic
Use sp to estimate unknown
, use a t test statisticwith
n1+ n22 d.f.
Use s1and s2 to estimateunknown 1and 2, use a t
test statisticand calculate the
required degrees of freedom
1and 2unknownbut assumed equal
1and 2unknown,not assumed equal
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2
2
2
1
2
1
2121
n
n
xxz
The test statistic for 12 is:
1and 2known
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Example 1
Pengeboran pertamina mempunyai 2 mesin
bor. Mesin Alpha mempunyai diameter 0.025
inch dan mesin Beta mempunyai 0.034 inch.
Apakah mesin Beta mempunyai rata2 yg lebihbesar drpd rata2 mesin Alpha. Jika 100 sample
dari kedua mesin diambil dan rata2 mesin aplha
0.501 inch dan mesin beta 0.509 inch
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Contoh 2
Page 444 no 10-21
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St H th i T t f T
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Steps: Hypothesis Test for Two
Population Means
1. Specify parameter of interest
2. Formulate hypotheses
3. Specify the significance level ()4. Construct the rejection region and develop the
decision rule
5. Compute the test statistics and/or the p-value6. Reach a decision
7. Draw a conclusionSee Chapter 9
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1and 2unknown,large samples
Where t has (n1+ n22) d.f.,and
2nn
s1ns1ns
21
2
22
2
11p
21
p
2121
n
1
n
1s
xx
t
The test statistic for 12 is:
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1and 2unknown,small samples
The test statistic for 12 is:
2
22
1
21
2121
n
s
n
s
xx
t
Where t has d.f. given by
1n
/ns
1n
/ns
)/ns/n(sdf
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1
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Two Population Means, Independent Samples
Lower tail test:
H0: 120HA: 12< 0
Upper tail test:
H0: 120
HA: 12> 0
Two-tailed test:
H0: 12= 0
HA: 120
/2 /2
-z
-z/2z z/2
Reject H0if z < -z Reject H0if z > z Reject H0if z < -z/2or z > z
/2
Hypothesis tests for 12
Example: 1and 2known:
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Example1and 2unknown, assumed equal
Youre a financial analyst for a brokerage firm. Is there a
difference in dividend yield between stocks listed on the
NYSE & NASDAQ? You collect the following data:
NYSE NASDAQ
Number 21 25
Sample mean 3.27 2.53
Sample std dev 1.30 1.16
Assuming equal variances, is
there 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
2
22
2
11p
2.040
251
2111.2256
02.533.27
n1
n1s
xxt
21
p
2121
The test statistic is:
Where:
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Solution
H0: 1 - 2= 0 i.e. (1 = 2)
HA: 1 - 20 i.e. (1 2)
= 0.05
df = 21 + 25 - 2 = 44Critical Values: t = 2.0154
Test Statistic: Decision:
Conclusion:Reject H0at = 0.05
There is evidence that
the means are different.
t0 2.0154-2.0154
0.025
Reject H0 Reject H0
0.025
2.040
2.040
25
1
21
11.2256
2.533.27t
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10.3 Paired Sample
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Paired Samples
Tests Means of 2 RelatedPopulations
Paired or matched samples e.g., two different value appraisals on the same home
Repeated measures (before/after) e.g., weight before and after a diet program
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, where
di= x1i- x2i
The point estimate forthe population mean
paired difference is d :
1n
)d(d
s
n
1i
2
i
d
n
d
d
n
1i
i
The sample standarddeviation is
n is the number of pairs in the paired sample
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Paired Differences
The confidence interval for d is
1n
)d(d
s
n
1i
2
i
d
nstd d
Where t has n - 1 d.f. and sd is:
(continued)
n is the number of pairs in the paired sample
H th i T ti f
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The test statistic for d is
1n
)d(d
s
n
1i
2
i
d
n
s dt dd
Where t has n - 1 d.f.
and sd is:
n is the
number
of pairs
in the
paired
sample
Hypothesis Testing forPaired Samples
H th i T ti f
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Lower tail test:
H0: d0HA: d< 0
Upper tail test:
H0: d0HA: d> 0
Two-tailed test:
H0: d= 0HA: d0
Paired Samples
Hypothesis Testing forPaired Samples
/2 /2
-t
-t/2t t/2
Reject H0if t < -t Reject H0if t > t Reject H0if t < -t/2or t > t
/2
Where t has n - 1 d.f.
(continued)
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Assume you send your salespeople to a customerservice training workshop. Is the training effective?
You collect the following data:
Paired Sample Example
Number of Complaints: (2) - (1)Salesperson Before (1) After (2) Difference,d
i
C.B. 6 4 - 2
T.F. 20 6 -14
M.H. 3 2 - 1
R.K. 0 0 0
M.O. 4 0 - 4
-21
d = din
5.67
1n
)d(ds
2
i
d
= -4.2
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Has the training made a difference in the number of complaints
(at the 0.05 level)?
- 4.2d =
1.6655.67/
04.2
n/s
dt
d
d
H0: d = 0
HA:
d0
Test Statistic:
Critical Value = 2.7765d.f. = n - 1 = 4
Reject
/2
- 2.7765 2.7765
Decision:Do not reject H0(t stat is not in the reject region)
Conclusion:There is not a
significant change in the
number of complaints.
Paired Sample: Solution
Reject
/2
- 1.66= 0.05
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10.4 Proportion
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Two Population Proportions
Goal: Form a confidence interval for
or test a hypothesis about the
difference between two population
proportions, 12
The point estimate for
the difference isp1p2
Assumptions:
n115 , n1(1-1) 5
n225 , n2(1-2) 5
Confidence Interval for
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Confidence Interval forTwo Population Proportions
222
1
11
21 n
)p(1p
n
)p(1p
zpp
The confidence interval for 12 is:
Hypothesis Tests for
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Hypothesis Tests forTwo Population Proportions
Population proportions
Lower tail test:
H0: 12HA: 1< 2
i.e.,
H0: 120HA: 12< 0
Upper tail test:
H0: 12HA: 1> 2
i.e.,
H0: 120HA: 12> 0
Two-tailed test:
H0: 1= 2HA: 12
i.e.,
H0: 12= 0HA: 120
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Two Population Proportions
21
21
21
2211
nn
xx
nn
pnpn
p
The pooled estimate for the
overall proportion is:
Where x1and x2are the numbers from
samples 1 and 2 with the characteristic of interest
Since we begin by assuming the null
hypothesis is true, we assume 1= 2
and pool the two p estimates
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Two Population Proportions
21
2121
n
1
n
1)p(1p
ppz
The test statistic for 12 is:
(continued)
Hypothesis Tests for
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Hypothesis Tests forTwo Population Proportions
Population proportions
Lower tail test:
H0: 120HA: 12< 0
Upper tail test:
H0: 120
HA: 12> 0
Two-tailed test:
H0: 12= 0
HA: 120
/2 /2
-z
-z/2z z/2
Reject H0if z < -z Reject H0if z > z Reject H0if z < -z/2or z > z
/2
Example:
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Example:Two population Proportions
Is there a significant difference between the
proportion of men and the proportion of
women who will vote Yes on Proposition A?
In a random sample, 36 of 72 men and 31 of
50 women indicated they would vote Yes
Test at the 0.05 level of significance
Example:
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The hypothesis test is:
H0: 12= 0 (the two proportions are equal)
HA: 120 (there is a significant difference between proportions)
The sample proportions are:
Men: p1= 36/72 = 0.50
Women: p2= 31/50 = 0.62
.5490122
67
5072
3136
nn
xxp
21
21
The pooled estimate for the overall proportion is:
Example:Two population Proportions
(continued)
Example:
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The test statistic for 12 is:
Example:Two population Proportions
(continued)
0.025
-1.96
1.96
0.025
-1.31
Decision:Do not reject H0
Conclusion:There is not
significant evidence of a
difference in the proportion
who will vote yes between
1.31
50
1
72
1.549)0(1.5490
0.620.500
n1
n1)p(1p
ppz
21
2121
Reject H0 Reject H0
Critical Values = 1.96For = 0.05