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Comparing Means from Two-Sample
Kwonsang Lee
University of Pennsylvania
kwonleewhartonupennedu
April 3 2015
Kwonsang Lee STAT111 April 3 2015 1 22
Inference from One-Sample
We have two options to make an inference about population mean micro fromone sample of size n1) 100(1-C) Confidence interval and2) Hypothesis test with a level α
1) 100(1-C) Confidence IntervalWe need to consider the case when σ is known or unknown
a Known σ(X minus Zlowast σradic
n X + Zlowast σradic
n)
b Unknown σ(X minus tlowastnminus1
sradicn X + tlowastnminus1
sradicn
)
Kwonsang Lee STAT111 April 3 2015 2 22
Inference from One-Sample
2) Hypothesis test with a level α
a State the null and alternative hypotheses (Here two-sided example)
H0 micro = micro0 and Ha micro 6= micro0
b Calculate a test statistic Z0 (known σ) or a test statistic T0 (unknownσ)
Z0 =X minus micro0
σradicn
or T0 =X minus micro0
sradicn
c Calculate the P-value
P-value =
2times P(Z ge |Z0|) σ is known
2times P(T ge |T0|) σ is unknown
d Compare the P-value to the significance level α
Kwonsang Lee STAT111 April 3 2015 3 22
Supplement of t-test (Two-sided test)
Because t-table doesnrsquot give the P-value we can modify our t-testInstead of computing P-value we can find the value tlowastnminus1 such that
P(T gt tlowastnminus1) =α
2
Then
Conclusion =
We reject the null if |T0| ge tlowastnminus1We donrsquot reject the null if |T0| lt tlowastnminus1
Note If one-sided alternative hypothesis is Ha micro gt 0 we need to find thevalue tlowastnminus1 such that P(T gt tlowastnminus1) = α We reject the null if T0 gt tlowastnminus1Also if Ha micro lt 0 we need to use tlowastnminus1 such that P(T lt tlowastnminus1) = α Wereject the null if T0 lt tlowastnminus1 Draw the t-distribution and think about it
Kwonsang Lee STAT111 April 3 2015 4 22
New Terminology Standard Error
X is from the population with mean micro and SD σ We take a sample ofsize n from the population and say X1 Xn
What we learned
A sample (X1 Xn) has the sample mean X and the sample SD s
The sample mean X has the distribution with mean micro and SD σradicn
New terminology
Standard error of X is sradicn
ie SE(X ) = sradicn
Kwonsang Lee STAT111 April 3 2015 5 22
Two-Sample Example
Letrsquos assume that we want to study about household incomes inPhiladelphia and New York
Philadelphia income dist rArr mean microp and SD σp
New York income dist rArr mean micron and SD σn
Then we take a Philadelphia sample of size np and a New York sample ofsize nn
Phila sample rArr sample mean xp and sample SD sp
NY sample rArr sample mean xn and sample SD sn
What to do We want to compare microp with micron
1) Hypothesis test of microp = micron
2) Confidence interval of microp minus micron
Kwonsang Lee STAT111 April 3 2015 6 22
Inference from Two-Sample Intro
We donrsquot know the values of micro1 and micro2 We want to make inferences fromSample 1 and Sample 2
We can conduct a hypothesis test or construct a confidence interval for microAlso we need to consider the case when σ1 and σ2 are known or unknown
1) Hypothesis test of micro1 = micro2 or micro1 minus micro2 = 0
2) Confidence interval of micro1 minus micro2
Kwonsang Lee STAT111 April 3 2015 7 22
Two-Sample Hypothesis Test Known σ1 and σ2
Since σ1 and σ2 are known we can take the Z test
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic Z0 is
Z0 =(X1 minus X2)minus (micro1 minus micro2)radic
σ21
n1+
σ22
n2
c P-value is
P(Z le minus|Z0|) + P(Z ge |Z0|) = 2times P(Z ge |Z0|)
d Compare P-value with a level α
Kwonsang Lee STAT111 April 3 2015 8 22
Two-Sample Hypothesis Test Unknown σ1 and σ2
Since σ1 and σ2 are unknown we use s1 and s2 instead and use the t-testwith a level α
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic T0 is
T0 =(X1 minus X2)minus (micro1 minus micro2)radic
s21n1
+s22n2
c (Modified Version) We can find the critical value tlowastk such that
P(T gt tlowastk ) =α
2
where k = min(n1 minus 1 n2 minus 1)
d Compare |T0| with the value tlowastk (|T0| gt tlowastk rArr Reject the null)
Kwonsang Lee STAT111 April 3 2015 9 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Inference from One-Sample
We have two options to make an inference about population mean micro fromone sample of size n1) 100(1-C) Confidence interval and2) Hypothesis test with a level α
1) 100(1-C) Confidence IntervalWe need to consider the case when σ is known or unknown
a Known σ(X minus Zlowast σradic
n X + Zlowast σradic
n)
b Unknown σ(X minus tlowastnminus1
sradicn X + tlowastnminus1
sradicn
)
Kwonsang Lee STAT111 April 3 2015 2 22
Inference from One-Sample
2) Hypothesis test with a level α
a State the null and alternative hypotheses (Here two-sided example)
H0 micro = micro0 and Ha micro 6= micro0
b Calculate a test statistic Z0 (known σ) or a test statistic T0 (unknownσ)
Z0 =X minus micro0
σradicn
or T0 =X minus micro0
sradicn
c Calculate the P-value
P-value =
2times P(Z ge |Z0|) σ is known
2times P(T ge |T0|) σ is unknown
d Compare the P-value to the significance level α
Kwonsang Lee STAT111 April 3 2015 3 22
Supplement of t-test (Two-sided test)
Because t-table doesnrsquot give the P-value we can modify our t-testInstead of computing P-value we can find the value tlowastnminus1 such that
P(T gt tlowastnminus1) =α
2
Then
Conclusion =
We reject the null if |T0| ge tlowastnminus1We donrsquot reject the null if |T0| lt tlowastnminus1
Note If one-sided alternative hypothesis is Ha micro gt 0 we need to find thevalue tlowastnminus1 such that P(T gt tlowastnminus1) = α We reject the null if T0 gt tlowastnminus1Also if Ha micro lt 0 we need to use tlowastnminus1 such that P(T lt tlowastnminus1) = α Wereject the null if T0 lt tlowastnminus1 Draw the t-distribution and think about it
Kwonsang Lee STAT111 April 3 2015 4 22
New Terminology Standard Error
X is from the population with mean micro and SD σ We take a sample ofsize n from the population and say X1 Xn
What we learned
A sample (X1 Xn) has the sample mean X and the sample SD s
The sample mean X has the distribution with mean micro and SD σradicn
New terminology
Standard error of X is sradicn
ie SE(X ) = sradicn
Kwonsang Lee STAT111 April 3 2015 5 22
Two-Sample Example
Letrsquos assume that we want to study about household incomes inPhiladelphia and New York
Philadelphia income dist rArr mean microp and SD σp
New York income dist rArr mean micron and SD σn
Then we take a Philadelphia sample of size np and a New York sample ofsize nn
Phila sample rArr sample mean xp and sample SD sp
NY sample rArr sample mean xn and sample SD sn
What to do We want to compare microp with micron
1) Hypothesis test of microp = micron
2) Confidence interval of microp minus micron
Kwonsang Lee STAT111 April 3 2015 6 22
Inference from Two-Sample Intro
We donrsquot know the values of micro1 and micro2 We want to make inferences fromSample 1 and Sample 2
We can conduct a hypothesis test or construct a confidence interval for microAlso we need to consider the case when σ1 and σ2 are known or unknown
1) Hypothesis test of micro1 = micro2 or micro1 minus micro2 = 0
2) Confidence interval of micro1 minus micro2
Kwonsang Lee STAT111 April 3 2015 7 22
Two-Sample Hypothesis Test Known σ1 and σ2
Since σ1 and σ2 are known we can take the Z test
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic Z0 is
Z0 =(X1 minus X2)minus (micro1 minus micro2)radic
σ21
n1+
σ22
n2
c P-value is
P(Z le minus|Z0|) + P(Z ge |Z0|) = 2times P(Z ge |Z0|)
d Compare P-value with a level α
Kwonsang Lee STAT111 April 3 2015 8 22
Two-Sample Hypothesis Test Unknown σ1 and σ2
Since σ1 and σ2 are unknown we use s1 and s2 instead and use the t-testwith a level α
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic T0 is
T0 =(X1 minus X2)minus (micro1 minus micro2)radic
s21n1
+s22n2
c (Modified Version) We can find the critical value tlowastk such that
P(T gt tlowastk ) =α
2
where k = min(n1 minus 1 n2 minus 1)
d Compare |T0| with the value tlowastk (|T0| gt tlowastk rArr Reject the null)
Kwonsang Lee STAT111 April 3 2015 9 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Inference from One-Sample
2) Hypothesis test with a level α
a State the null and alternative hypotheses (Here two-sided example)
H0 micro = micro0 and Ha micro 6= micro0
b Calculate a test statistic Z0 (known σ) or a test statistic T0 (unknownσ)
Z0 =X minus micro0
σradicn
or T0 =X minus micro0
sradicn
c Calculate the P-value
P-value =
2times P(Z ge |Z0|) σ is known
2times P(T ge |T0|) σ is unknown
d Compare the P-value to the significance level α
Kwonsang Lee STAT111 April 3 2015 3 22
Supplement of t-test (Two-sided test)
Because t-table doesnrsquot give the P-value we can modify our t-testInstead of computing P-value we can find the value tlowastnminus1 such that
P(T gt tlowastnminus1) =α
2
Then
Conclusion =
We reject the null if |T0| ge tlowastnminus1We donrsquot reject the null if |T0| lt tlowastnminus1
Note If one-sided alternative hypothesis is Ha micro gt 0 we need to find thevalue tlowastnminus1 such that P(T gt tlowastnminus1) = α We reject the null if T0 gt tlowastnminus1Also if Ha micro lt 0 we need to use tlowastnminus1 such that P(T lt tlowastnminus1) = α Wereject the null if T0 lt tlowastnminus1 Draw the t-distribution and think about it
Kwonsang Lee STAT111 April 3 2015 4 22
New Terminology Standard Error
X is from the population with mean micro and SD σ We take a sample ofsize n from the population and say X1 Xn
What we learned
A sample (X1 Xn) has the sample mean X and the sample SD s
The sample mean X has the distribution with mean micro and SD σradicn
New terminology
Standard error of X is sradicn
ie SE(X ) = sradicn
Kwonsang Lee STAT111 April 3 2015 5 22
Two-Sample Example
Letrsquos assume that we want to study about household incomes inPhiladelphia and New York
Philadelphia income dist rArr mean microp and SD σp
New York income dist rArr mean micron and SD σn
Then we take a Philadelphia sample of size np and a New York sample ofsize nn
Phila sample rArr sample mean xp and sample SD sp
NY sample rArr sample mean xn and sample SD sn
What to do We want to compare microp with micron
1) Hypothesis test of microp = micron
2) Confidence interval of microp minus micron
Kwonsang Lee STAT111 April 3 2015 6 22
Inference from Two-Sample Intro
We donrsquot know the values of micro1 and micro2 We want to make inferences fromSample 1 and Sample 2
We can conduct a hypothesis test or construct a confidence interval for microAlso we need to consider the case when σ1 and σ2 are known or unknown
1) Hypothesis test of micro1 = micro2 or micro1 minus micro2 = 0
2) Confidence interval of micro1 minus micro2
Kwonsang Lee STAT111 April 3 2015 7 22
Two-Sample Hypothesis Test Known σ1 and σ2
Since σ1 and σ2 are known we can take the Z test
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic Z0 is
Z0 =(X1 minus X2)minus (micro1 minus micro2)radic
σ21
n1+
σ22
n2
c P-value is
P(Z le minus|Z0|) + P(Z ge |Z0|) = 2times P(Z ge |Z0|)
d Compare P-value with a level α
Kwonsang Lee STAT111 April 3 2015 8 22
Two-Sample Hypothesis Test Unknown σ1 and σ2
Since σ1 and σ2 are unknown we use s1 and s2 instead and use the t-testwith a level α
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic T0 is
T0 =(X1 minus X2)minus (micro1 minus micro2)radic
s21n1
+s22n2
c (Modified Version) We can find the critical value tlowastk such that
P(T gt tlowastk ) =α
2
where k = min(n1 minus 1 n2 minus 1)
d Compare |T0| with the value tlowastk (|T0| gt tlowastk rArr Reject the null)
Kwonsang Lee STAT111 April 3 2015 9 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Supplement of t-test (Two-sided test)
Because t-table doesnrsquot give the P-value we can modify our t-testInstead of computing P-value we can find the value tlowastnminus1 such that
P(T gt tlowastnminus1) =α
2
Then
Conclusion =
We reject the null if |T0| ge tlowastnminus1We donrsquot reject the null if |T0| lt tlowastnminus1
Note If one-sided alternative hypothesis is Ha micro gt 0 we need to find thevalue tlowastnminus1 such that P(T gt tlowastnminus1) = α We reject the null if T0 gt tlowastnminus1Also if Ha micro lt 0 we need to use tlowastnminus1 such that P(T lt tlowastnminus1) = α Wereject the null if T0 lt tlowastnminus1 Draw the t-distribution and think about it
Kwonsang Lee STAT111 April 3 2015 4 22
New Terminology Standard Error
X is from the population with mean micro and SD σ We take a sample ofsize n from the population and say X1 Xn
What we learned
A sample (X1 Xn) has the sample mean X and the sample SD s
The sample mean X has the distribution with mean micro and SD σradicn
New terminology
Standard error of X is sradicn
ie SE(X ) = sradicn
Kwonsang Lee STAT111 April 3 2015 5 22
Two-Sample Example
Letrsquos assume that we want to study about household incomes inPhiladelphia and New York
Philadelphia income dist rArr mean microp and SD σp
New York income dist rArr mean micron and SD σn
Then we take a Philadelphia sample of size np and a New York sample ofsize nn
Phila sample rArr sample mean xp and sample SD sp
NY sample rArr sample mean xn and sample SD sn
What to do We want to compare microp with micron
1) Hypothesis test of microp = micron
2) Confidence interval of microp minus micron
Kwonsang Lee STAT111 April 3 2015 6 22
Inference from Two-Sample Intro
We donrsquot know the values of micro1 and micro2 We want to make inferences fromSample 1 and Sample 2
We can conduct a hypothesis test or construct a confidence interval for microAlso we need to consider the case when σ1 and σ2 are known or unknown
1) Hypothesis test of micro1 = micro2 or micro1 minus micro2 = 0
2) Confidence interval of micro1 minus micro2
Kwonsang Lee STAT111 April 3 2015 7 22
Two-Sample Hypothesis Test Known σ1 and σ2
Since σ1 and σ2 are known we can take the Z test
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic Z0 is
Z0 =(X1 minus X2)minus (micro1 minus micro2)radic
σ21
n1+
σ22
n2
c P-value is
P(Z le minus|Z0|) + P(Z ge |Z0|) = 2times P(Z ge |Z0|)
d Compare P-value with a level α
Kwonsang Lee STAT111 April 3 2015 8 22
Two-Sample Hypothesis Test Unknown σ1 and σ2
Since σ1 and σ2 are unknown we use s1 and s2 instead and use the t-testwith a level α
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic T0 is
T0 =(X1 minus X2)minus (micro1 minus micro2)radic
s21n1
+s22n2
c (Modified Version) We can find the critical value tlowastk such that
P(T gt tlowastk ) =α
2
where k = min(n1 minus 1 n2 minus 1)
d Compare |T0| with the value tlowastk (|T0| gt tlowastk rArr Reject the null)
Kwonsang Lee STAT111 April 3 2015 9 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
New Terminology Standard Error
X is from the population with mean micro and SD σ We take a sample ofsize n from the population and say X1 Xn
What we learned
A sample (X1 Xn) has the sample mean X and the sample SD s
The sample mean X has the distribution with mean micro and SD σradicn
New terminology
Standard error of X is sradicn
ie SE(X ) = sradicn
Kwonsang Lee STAT111 April 3 2015 5 22
Two-Sample Example
Letrsquos assume that we want to study about household incomes inPhiladelphia and New York
Philadelphia income dist rArr mean microp and SD σp
New York income dist rArr mean micron and SD σn
Then we take a Philadelphia sample of size np and a New York sample ofsize nn
Phila sample rArr sample mean xp and sample SD sp
NY sample rArr sample mean xn and sample SD sn
What to do We want to compare microp with micron
1) Hypothesis test of microp = micron
2) Confidence interval of microp minus micron
Kwonsang Lee STAT111 April 3 2015 6 22
Inference from Two-Sample Intro
We donrsquot know the values of micro1 and micro2 We want to make inferences fromSample 1 and Sample 2
We can conduct a hypothesis test or construct a confidence interval for microAlso we need to consider the case when σ1 and σ2 are known or unknown
1) Hypothesis test of micro1 = micro2 or micro1 minus micro2 = 0
2) Confidence interval of micro1 minus micro2
Kwonsang Lee STAT111 April 3 2015 7 22
Two-Sample Hypothesis Test Known σ1 and σ2
Since σ1 and σ2 are known we can take the Z test
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic Z0 is
Z0 =(X1 minus X2)minus (micro1 minus micro2)radic
σ21
n1+
σ22
n2
c P-value is
P(Z le minus|Z0|) + P(Z ge |Z0|) = 2times P(Z ge |Z0|)
d Compare P-value with a level α
Kwonsang Lee STAT111 April 3 2015 8 22
Two-Sample Hypothesis Test Unknown σ1 and σ2
Since σ1 and σ2 are unknown we use s1 and s2 instead and use the t-testwith a level α
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic T0 is
T0 =(X1 minus X2)minus (micro1 minus micro2)radic
s21n1
+s22n2
c (Modified Version) We can find the critical value tlowastk such that
P(T gt tlowastk ) =α
2
where k = min(n1 minus 1 n2 minus 1)
d Compare |T0| with the value tlowastk (|T0| gt tlowastk rArr Reject the null)
Kwonsang Lee STAT111 April 3 2015 9 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Two-Sample Example
Letrsquos assume that we want to study about household incomes inPhiladelphia and New York
Philadelphia income dist rArr mean microp and SD σp
New York income dist rArr mean micron and SD σn
Then we take a Philadelphia sample of size np and a New York sample ofsize nn
Phila sample rArr sample mean xp and sample SD sp
NY sample rArr sample mean xn and sample SD sn
What to do We want to compare microp with micron
1) Hypothesis test of microp = micron
2) Confidence interval of microp minus micron
Kwonsang Lee STAT111 April 3 2015 6 22
Inference from Two-Sample Intro
We donrsquot know the values of micro1 and micro2 We want to make inferences fromSample 1 and Sample 2
We can conduct a hypothesis test or construct a confidence interval for microAlso we need to consider the case when σ1 and σ2 are known or unknown
1) Hypothesis test of micro1 = micro2 or micro1 minus micro2 = 0
2) Confidence interval of micro1 minus micro2
Kwonsang Lee STAT111 April 3 2015 7 22
Two-Sample Hypothesis Test Known σ1 and σ2
Since σ1 and σ2 are known we can take the Z test
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic Z0 is
Z0 =(X1 minus X2)minus (micro1 minus micro2)radic
σ21
n1+
σ22
n2
c P-value is
P(Z le minus|Z0|) + P(Z ge |Z0|) = 2times P(Z ge |Z0|)
d Compare P-value with a level α
Kwonsang Lee STAT111 April 3 2015 8 22
Two-Sample Hypothesis Test Unknown σ1 and σ2
Since σ1 and σ2 are unknown we use s1 and s2 instead and use the t-testwith a level α
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic T0 is
T0 =(X1 minus X2)minus (micro1 minus micro2)radic
s21n1
+s22n2
c (Modified Version) We can find the critical value tlowastk such that
P(T gt tlowastk ) =α
2
where k = min(n1 minus 1 n2 minus 1)
d Compare |T0| with the value tlowastk (|T0| gt tlowastk rArr Reject the null)
Kwonsang Lee STAT111 April 3 2015 9 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Inference from Two-Sample Intro
We donrsquot know the values of micro1 and micro2 We want to make inferences fromSample 1 and Sample 2
We can conduct a hypothesis test or construct a confidence interval for microAlso we need to consider the case when σ1 and σ2 are known or unknown
1) Hypothesis test of micro1 = micro2 or micro1 minus micro2 = 0
2) Confidence interval of micro1 minus micro2
Kwonsang Lee STAT111 April 3 2015 7 22
Two-Sample Hypothesis Test Known σ1 and σ2
Since σ1 and σ2 are known we can take the Z test
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic Z0 is
Z0 =(X1 minus X2)minus (micro1 minus micro2)radic
σ21
n1+
σ22
n2
c P-value is
P(Z le minus|Z0|) + P(Z ge |Z0|) = 2times P(Z ge |Z0|)
d Compare P-value with a level α
Kwonsang Lee STAT111 April 3 2015 8 22
Two-Sample Hypothesis Test Unknown σ1 and σ2
Since σ1 and σ2 are unknown we use s1 and s2 instead and use the t-testwith a level α
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic T0 is
T0 =(X1 minus X2)minus (micro1 minus micro2)radic
s21n1
+s22n2
c (Modified Version) We can find the critical value tlowastk such that
P(T gt tlowastk ) =α
2
where k = min(n1 minus 1 n2 minus 1)
d Compare |T0| with the value tlowastk (|T0| gt tlowastk rArr Reject the null)
Kwonsang Lee STAT111 April 3 2015 9 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Two-Sample Hypothesis Test Known σ1 and σ2
Since σ1 and σ2 are known we can take the Z test
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic Z0 is
Z0 =(X1 minus X2)minus (micro1 minus micro2)radic
σ21
n1+
σ22
n2
c P-value is
P(Z le minus|Z0|) + P(Z ge |Z0|) = 2times P(Z ge |Z0|)
d Compare P-value with a level α
Kwonsang Lee STAT111 April 3 2015 8 22
Two-Sample Hypothesis Test Unknown σ1 and σ2
Since σ1 and σ2 are unknown we use s1 and s2 instead and use the t-testwith a level α
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic T0 is
T0 =(X1 minus X2)minus (micro1 minus micro2)radic
s21n1
+s22n2
c (Modified Version) We can find the critical value tlowastk such that
P(T gt tlowastk ) =α
2
where k = min(n1 minus 1 n2 minus 1)
d Compare |T0| with the value tlowastk (|T0| gt tlowastk rArr Reject the null)
Kwonsang Lee STAT111 April 3 2015 9 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Two-Sample Hypothesis Test Unknown σ1 and σ2
Since σ1 and σ2 are unknown we use s1 and s2 instead and use the t-testwith a level α
a H0 micro1 minus micro2 = 0 and Ha micro1 minus micro2 6= 0
b Test statistic T0 is
T0 =(X1 minus X2)minus (micro1 minus micro2)radic
s21n1
+s22n2
c (Modified Version) We can find the critical value tlowastk such that
P(T gt tlowastk ) =α
2
where k = min(n1 minus 1 n2 minus 1)
d Compare |T0| with the value tlowastk (|T0| gt tlowastk rArr Reject the null)
Kwonsang Lee STAT111 April 3 2015 9 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Example 1
There is a product A that is advertised as helping students to learnStatistics more effectively We want to test if there is any positive effect ofthe product A Among 44 participants we randomly select 21 people touse the product (21 treated and 23 control) After one month allparticipants take a statistic test and the scores are recorded Thefollowing is the summary
n x s
Treated 21 515 11Control 23 415 17
Q How can we conduct a hypothesis test
We need to do Two-Sample t-test
Kwonsang Lee STAT111 April 3 2015 10 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Example 1
Two-Sample t-test with a level α = 005
a H0 microtreated minus microcontrol = 0 and Ha microt minus microc 6= 0
b Test statistic T0 is given by
T0 =(Xt minus Xc)minus (microt minus microc)radic
s2tnt
+ s2cnc
=(515minus 415)minus 0radic
112
21 + 172
23
= 2336
c Conservatively degree of freedom k is min(nt minus 1 nc minus 1) = 20 Thecritical value tlowastk is 2086
P(T gt tlowast20) =α
2= 0025
d Since |T0| = 2336 gt 2086 = tlowast20 we reject the null hypothesis
t-table rArr http
bcswhfreemancomips6econtentcat_050ips6e_table-dpdf
Kwonsang Lee STAT111 April 3 2015 11 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Two-Sample t-test in JMP
Here are the references for t-test
One-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_one_sample_t-testpdf
Two-sample t-test httpwwwchemscedufacultymorganresourcesstatisticsJMP_two_sample_t-testpdf
Steps for two-sample t-test
1 Open the data file
2 Go to Analyze rarr Fit Y by X For example Y is a score variable andX is an indicator of either lsquotreatedrsquo or lsquocontrolrsquo
3 Click the red triangle next to lsquoOneway Analysis of rsquo and chooset-test
Kwonsang Lee STAT111 April 3 2015 12 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Using JMP
We can find the descriptions of each sample We also find a confidenceinterval of micro1 minus micro2 and the results of Two-sample t-test
Using JMP we can compute the p-value of our test statistic T0 in theprevious Example 1 The P-value is 00264 which is less than 005 so wereject the null hypothesis
Here is another reference relate with Example 1httpwebutkedu~cwiek201TutorialsTwoSampleTtest
Kwonsang Lee STAT111 April 3 2015 13 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Confidence Interval from Two-Sample
We can consider two cases 1) Known σ1 and σ2 case and 2) Unknown σ1and σ2 case
Confidence interval of lsquomicro1 minus micro2rsquo with known σ1 and σ2
(X1 minus X2)plusmn Z lowast
radicσ21n1
+σ22n2
Confidence interval of lsquomicro1 minus micro2rsquo with unknown σ1 and σ2
(X1 minus X2)plusmn tlowastk
radics21n1
+s22n2
where k = min(n1 minus 1 n2 minus 1)
Kwonsang Lee STAT111 April 3 2015 14 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Special Case Matched Pairs
Sometimes the two samples that are being compared are matched pairs
For example if there is a drug A and it can lower blood pressure Eachsubjectrsquos blood pressure is measured before taking the drug and ismeasured after intake One subject has two values of the outcome Thenwe want to test if there is any difference between blood pressure beforeintake and blood pressure after intake
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
We want to test if blood pressure before = blood pressure after
Kwonsang Lee STAT111 April 3 2015 15 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Matched Pairs
In this case we can compute the difference D = X1 minus X2 HereDiff=BeforeminusAfter
Subject 1 Subject 2 Subject n
Before 130 128 126After 116 110 108
Diff 14 18 18
Then we can use a test like H0 Diff = 0 This is lsquoOne-Sample t-testrsquo
Kwonsang Lee STAT111 April 3 2015 16 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Matched Pairs Test
From Matched pairs design we have X1 and X2 for n subjects Then wecompute the new variable D = X1 minus X2 and compute the sample meanand the sample SD of D Xd and sd
Then we can state the null hypothesis H0 microd = 0 and the alternativeHa microd 6= 0 We calculate the test statistic T0
T0 =Xd minus microdsdradicn
Then we can calculate the critical value tlowastnminus1 and compare it with the teststatistic T0
Kwonsang Lee STAT111 April 3 2015 17 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Example 2
We consider the drug of lowering blood pressure example
Subject Before After D
1 130 116 142 128 110 18
10 126 108 18
The summary is that
xbefore = 1222 sbefore = 63
xafter = 113 safter = 91
However in matched pairs designwhat we need is a new variableD = Before minus After We have
xd = 92 sd = 98
Kwonsang Lee STAT111 April 3 2015 18 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Example 2 Under Independent Assumption
It is clear that lsquoBeforersquo and lsquoAfterrsquo is not independent because these twoare from the same subject If we consider lsquoBeforersquo and lsquoAfterrsquo areindependent then what can we conclude
We want to do hypothesis test with a level α = 002
a H0 microbefore minus microafter = 0 and Ha microbefore minus microafter 6= 0
b The test statistic T0 is
T0 =(xbefore minus xafter )minus (microbefore minus microafter )radic
s21n1
+s22n2
=1222minus 113radic
632
10 + 912
10
= 2629
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2629 lt 2821 = tlowast9 So we donrsquot reject the null It meansthat there is not enough evidence that there is an effect of a drug onlowering blood pressure
Kwonsang Lee STAT111 April 3 2015 19 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Example 2 Matched Pairs
lsquoBeforersquo and lsquoAfterrsquo from the original data are dependent So doing t-testisnrsquot correct under independence Here is right Matched pairs t-test
Hypothesis test with a level α = 002
a H0 microd = 0 and Ha microd 6= 0
b The test statistic T0 is
T0 =xd minus microdsdradicn
=92
98radic
10= 2969
c k = n minus 1 = 9 and tlowast9 = 2821 such that P(T gt tlowast9 ) = α2 = 001
d |T0| = 2969 gt 2821 = tlowast9 So we reject the null It means thatthere is enough evidence that there is an effect of a drug on loweringblood pressure
Note It is important to use the right approach
Kwonsang Lee STAT111 April 3 2015 20 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Summary
We learned CI and Hypothesis test in so many situations I can give adirection about how to choose the right way of analysis
1 Need to understand the data ie is it one-sample or two-sampleor matched pairs design
2 Do we know the population SD σ
3 Is our goal making a CI or doing hypothesis test
4 If we need to do hypothesis test what is the null and alternativehypotheses (One-sided or Two-sided)
Kwonsang Lee STAT111 April 3 2015 21 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
Next Week
We have been talking about inferences of the population mean micro Nextweek wersquore going to talk about CI and hypothesis test for population p
Kwonsang Lee STAT111 April 3 2015 22 22
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