two sample tests test for equal variances test for equal means hhohho hhahha population 1 population...
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Two Sample TestsTwo Sample TestsTEST FOR EQUAL VARIANCESTEST FOR EQUAL VARIANCES TEST FOR EQUAL MEANSTEST FOR EQUAL MEANS
HHo
HHa
Population 1
Population 2
Population 1
Population 2
HHo
HHa
Population 1
Population 2
Population 1 Population 2
Hypothesis Tests for Two Hypothesis Tests for Two Population VariancesPopulation Variances
Two-Tailed Two-Tailed TestTest
Upper One-Upper One-Tailed TestTailed Test
Lower One-Lower One-Tailed TestTailed Test
22
21
22
210
:
:
AH
H22
21
22
210
:
:
AH
H22
21
22
210
:
:
AH
H
Hypothesis Tests for Two Hypothesis Tests for Two Population VariancesPopulation Variances
F-TEST STATISTIC FOR TESTING WHETHER F-TEST STATISTIC FOR TESTING WHETHER TWO POPULATIONS HAVE EQUAL TWO POPULATIONS HAVE EQUAL
VARIANCESVARIANCES
where:ni = Sample size from ith population
nj = Sample size from jth population
si2= Sample variance from ith
populationsj
2= Sample variance from jth population
)( 12112
2
jij
i nDandnDdfs
sF
Hypothesis Tests for Two Hypothesis Tests for Two Population VariancesPopulation Variances
(Example 9-2)(Example 9-2)
F 0
df: Di = 10, Dj =12
Rejection Region /2 = 0.05
76.22/ F
47.1017.0
025.022
21 s
sF
Since F=1.47 F/2= 2.76, do not reject H0
22
21
22
210
:
:
AH
H
= .10
F = 1.47
Independent SamplesIndependent Samples
Independent samplesIndependent samples Selected from two or more populationsValues in one sample have no influence on the values in the other sample(s).
Hypothesis Tests for Two Hypothesis Tests for Two Population MeansPopulation Means
Format 1Format 1Two-Tailed Two-Tailed TestTest
Upper One-Upper One-Tailed TestTailed Test
Lower One-Lower One-Tailed TestTailed Test
0.0:
0.0:
21
210
AH
H
0.0:
0.0:
21
210
AH
H
0.0:
0.0:
21
210
AH
H
Hypothesis Tests for Two Hypothesis Tests for Two Population MeansPopulation Means
Format 2Format 2Two-Tailed Two-Tailed TestTest
Upper One-Upper One-Tailed TestTailed Test
Lower One-Lower One-Tailed TestTailed Test
21
210
:
:
AH
H
21
210
:
:
AH
H
21
210
:
:
AH
H
Hypothesis Tests for Two Hypothesis Tests for Two Population MeansPopulation Means
T-TEST STATISTIC T-TEST STATISTIC
(EQUAL POPULATION VARIANCES)(EQUAL POPULATION VARIANCES)
where:Sample means from populations
1 and 2Hypothesized differenceSample sizes from the two
populationsPooled standard deviation
:21 xandx
21̀ :21 nandn
ps
21
2121
11
)()(
nns
xxt
p
221 nndf
Hypothesis Tests for Two Hypothesis Tests for Two Population MeansPopulation Means
POOLED STANDARD DEVIATIONPOOLED STANDARD DEVIATION
Where:
s12 = Sample variance from
population 1s2
2 = Sample variance from population 2n1 and n2 = Sample sizes from populations 1 and 2 respectively
2
)1()1(
21
222
211
nn
snsnsp
Hypothesis Tests for Two Hypothesis Tests for Two Population MeansPopulation Means
t-TEST STATISTICt-TEST STATISTIC
where:s1
2 = Sample variance from population 1
s22 = Sample variance from
population 2
(Unequal Variances)
2
22
1
21
2121 )(
ns
ns
xxt
048.22/ t0
Hypothesis Tests for Two Hypothesis Tests for Two Population Means Population Means
(Example 9-4)(Example 9-4)
Rejection Region /2 = 0.025
Since t < 2.048, do not reject H0
048.22/ t
Rejection Region /2 = 0.025
465.0
151
151
23.677
)0.0()140.2255.2()(
2
22
1
21
2121
ns
ns
xxt
0.0:
0.0:
21
210
AH
H
Hypothesis Tests for Two Hypothesis Tests for Two Population MeansPopulation Means
DEGREES OF FREEDOM FOR t-TEST DEGREES OF FREEDOM FOR t-TEST STATISTIC WITH UNEQUAL POPULATION STATISTIC WITH UNEQUAL POPULATION
VARIANCESVARIANCES
)1)/(
1)/(
(
)//(
2
22
22
1
21
21
2221
21
nns
nns
nsns
Confidence Interval Confidence Interval Estimates for Estimates for 11 - - 22
STANDARD DEVIATIONS UNKNOWN STANDARD DEVIATIONS UNKNOWN
ANDAND 1122 = = 22
22
where: = Pooled standard
deviation
t/2 = critical value from t-distribution for desired confidence level and degrees of freedom equal to n1 + n2 -2
212/21
11)(
nnstxx p
2
)1()1(
21
222
211
nn
snsnsp
Confidence Interval Confidence Interval Estimates for Estimates for 11 - - 22
(Example 9-5)(Example 9-5)
94.563$)45.508,7$39.072,8($21 xx
72.086,1299
)11.813)(19()12.304,1)(19(
2
)1()1( 22
21
222
211
nn
snsnsp
40.894$94.563$9
1
9
1)72.086,1)(7459.1(94.563$
- - $330.4$330.4
66
$1,458.$1,458.3434
Confidence Interval Confidence Interval Estimates for Estimates for 11 - - 22
STANDARD DEVIATIONS UNKNOWN STANDARD DEVIATIONS UNKNOWN
ANDAND 1122 22
22
where:t/2 = critical value from t-distribution for desired confidence level
and degrees of freedom equal to:
2
22
1
21
2/21 )(n
s
n
stxx
)1)/(
1)/(
(
)//(
2
22
22
1
21
21
2221
21
nns
nns
nsns
Confidence Interval Confidence Interval Estimates for Estimates for 11 - - 22
LARGE SAMPLE SIZESLARGE SAMPLE SIZES
where:z/2 = critical value from the standard
normal distribution for desired confidence level
2
22
1
21
2/21 )(n
s
n
szxx
Paired Samples Paired Samples Hypothesis Testing and Hypothesis Testing and
EstimationEstimation
Paired samplesPaired samples are samples that selected such that each data value from one sample is related (or matched) with a corresponding data value from the second sample. The sample values from one population have the potential to influence the probability that values will be selected from the second population.
Paired Samples Paired Samples Hypothesis Testing and Hypothesis Testing and
EstimationEstimation
PAIRED DIFFERENCEPAIRED DIFFERENCE
where: d = Paired difference
x1 and x2 = Values from sample 1 and 2, respectively
21 xxd
Paired Samples Paired Samples Hypothesis Testing and Hypothesis Testing and
EstimationEstimation
MEAN PAIRED DIFFERENCEMEAN PAIRED DIFFERENCE
where: di = ith paired difference
n = Number of paired differences
n
iidd
1
Paired Samples Paired Samples Hypothesis Testing and Hypothesis Testing and
EstimationEstimation
STANDARD DEVIATION FOR PAIRED STANDARD DEVIATION FOR PAIRED DIFFERENCESDIFFERENCES
where: di = ith paired difference
= Mean paired difference
1
)(1
2
n
dds
n
ii
d
d
Paired Samples Paired Samples Hypothesis Testing and Hypothesis Testing and
EstimationEstimation
t-TEST STATISTIC FOR PAIRED t-TEST STATISTIC FOR PAIRED DIFFERENCESDIFFERENCES
where: = Mean paired difference
d = Hypothesized paired difference
sd = Sample standard deviation of paired differences
n = Number of paired differences
d
1
ndf
n
sd
td
d
833.1t0
Paired Samples Paired Samples Hypothesis Testing and Hypothesis Testing and
EstimationEstimation(Example 9-6)(Example 9-6)
Rejection Region = 0.05
Since t=0.9165 < 1.833, do not reject H0
05.0
0.1:
0.1:0
dA
d
H
H
9165.0
10
382.40.127.2
n
sd
td
d
Paired Samples Paired Samples Hypothesis Testing and Hypothesis Testing and
EstimationEstimation
PAIRED CONFIDENCE INTERVAL PAIRED CONFIDENCE INTERVAL ESTIMATEESTIMATE
n
std d
2/
Paired Samples Paired Samples Hypothesis Testing and Hypothesis Testing and
EstimationEstimation(Example 9-7)(Example 9-7)
95% Confidence Interval
173.210.7100
95.109842.110.7
2/
n
std d
4.9274.927 9.2739.273
Hypothesis Tests for Two Hypothesis Tests for Two Population ProportionsPopulation Proportions
Format 1Format 1Two-Tailed Two-Tailed TestTest
Upper One-Upper One-Tailed TestTailed Test
Lower One-Lower One-Tailed TestTailed Test
0.0:
0.0:
21
210
AH
H
0.0:
0.0:
21
210
AH
H
0.0:
0.0:
21
210
AH
H
Hypothesis Tests for Two Hypothesis Tests for Two Population ProportionsPopulation Proportions
Format 2Format 2Two-Tailed Two-Tailed TestTest
Upper One-Upper One-Tailed TestTailed Test
Lower One-Lower One-Tailed TestTailed Test
21
210
:
:
AH
H
21
210
:
:
AH
H
21
210
:
:
AH
H
Hypothesis Tests for Two Hypothesis Tests for Two Population ProportionsPopulation Proportions
POOLED ESTIMATOR FOR OVERALL POOLED ESTIMATOR FOR OVERALL PROPORTIONPROPORTION
where: x1 and x2 = number from samples 1 and 2 with desired characteristic.
21
21
21
2211
nn
xx
nn
pnpnp
Hypothesis Tests for Two Hypothesis Tests for Two Population ProportionsPopulation Proportions
TEST STATISTIC FOR DIFFERENCE IN TEST STATISTIC FOR DIFFERENCE IN POPULATION PROPORTIONSPOPULATION PROPORTIONS
where: (1 - 2) = Hypothesized difference in proportions
from populations 1 and 2, respectivelyp1 and p2 = Sample proportions for samples selected from population 1 and 2
= Pooled estimator for the overall proportion for both populations combined
)11
)(1(
)()(
21
2121
nnpp
ppz
p
645.1z 0
Hypothesis Tests for Two Hypothesis Tests for Two Population Proportions Population Proportions
(Example 9-8)(Example 9-8)
Rejection Region = 0.05
Since z =-2.04 < -1.645, reject H0
75,55
250,250
0:
0:
21
21
21
210
xx
nn
H
H
A
04.2
)2501
2501
)(26.01(26.0
0)30.022.0(
)11
)(1(
)()(
21
2121
nnpp
ppz
Confidence Intervals for Confidence Intervals for Two Population Two Population
ProportionsProportions
CONFIDENCE INTERVAL ESTIMATE CONFIDENCE INTERVAL ESTIMATE
FOR FOR 11- - 22
where:p1 = Sample proportion from population 1
p2 = Sample proportion from population 2
z = Critical value from the standard normal table
2
22
1
112/21
)1(()1(()(
n
pp
n
ppzpp
Confidence Intervals for Confidence Intervals for Two Population Two Population
ProportionsProportions(Example 9-10)(Example 9-10)
w
ww
m
mmwm n
pp
n
ppzpp
)1(()1(()( 2/
530.0370
196565.0
425
240 wm pandp
069.0035.0370
)530.01)(530.0(
425
)565.01)(565.0(96.1)530.0565.0(
--0.030.0344
0.100.1044
Key TermsKey Terms
Independent Samples
Paired Samples