Lecture 10: Chapter 6: t-Test | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

T-TEST. For example, two methods for making a chemical analysis are compared to see if the new one is equivalent to the older standard method; algae are grown under different conditions to study a factor that is thought to stimulate growth; etc. “Do two different methods of doing A give different results?”. “Can we be highly confident that the difference is positive or negative?” . “How large might the difference be?” Tài liệu giúp bạn tham khảo, ôn tập và đạt kết quả cao. Mời bạn đọc đón xem!

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Môn: Applied statistics (ENEE1006IU)

Trường: Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

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Profile Nhi Nhi
lOMoARcPSD| 45903860
APPLIED STATISTICS
COURSE CODE: ENEE1006IU
Lecture 10:
Chapter 6: t-Test
(3 credits: 2 is for lecture, 1 is for lab-work)
1
lOMoARcPSD| 45903860
T-TEST
• For example, two methods for making a chemical analysis are compared to see if the
new one is equivalent to the older standard method; algae are grown under dierent
condions to study a factor that is thought to smulate growth; etc.
“Do two dierent methods of doing A give dierent results?”
“Can we be highly condent that the dierence is posive or negave?”
“How large might the dierence be?”
• One experimental design is to make a series of tests using treatment A and then to
independently make a series of tests using method B. Independent t-test
• A second way of designing the experiment is to pair the samples according to me,
technician, batch of material, or other factors that might contribute to a dierence
between the two measurements Paired t-test
lOMoARcPSD| 45903860
3
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES
•Two samples are said to be paired when each data point in the rst sample is
matched and related to a unique data point in the second sample.
•Paired experiments are used when it is dicult to control all factors that might
inuence the outcome.
•If these factors cannot be controlled, the experiment is arranged so they are
equally likely to inuence both of the paired observaons. Paired data are
evaluated using the paired t-test •The classical null hypothesis is:
“The dierence between the two methods is zero.”
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES
•The paired t-test examines the average of the dierences between paired
observaons
lOMoARcPSD| 45903860
•Let (X
11
, X
21
), (X
12
, X
22
), … , (X
1n
, X
2n
) be a set of n paired observaons where:
-µ
1
and
are the mean and variance of the populaon represented by X
1
-µ
2
and are the mean and variance of the populaon represented by X
2
Dene the dierences between each pair of observaons as D
j
=X
1j
- X
2j
, (j= 1, 2, …,
n)
The D
j
’s are assumed to be normally distributed with mean µ
D
and variance
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES
lOMoARcPSD| 45903860
5
: the sample average of the n dierences D
1
, D
2
, … , D
n
S
D
: the
sample standard deviaon of these dierences
:
:
:
lOMoARcPSD| 45903860
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES
H
a
:
H
a
:
H
a
:
lOMoARcPSD| 45903860
7
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES
lOMoARcPSD| 45903860
End of le 1.
Any quesons?
8
6.2. INDEPENDENT T-TEST FOR ASSESSING THE DIFFERENCE OF TWO AVERAGES
•A t-test on the dierence of the averages would conclude that A and B are not
dierent.
•Somemes it is not possible to pair the tests, and then the averages of the two
treatments must be compared using the independent t-test.
Example:
lOMoARcPSD| 45903860
•Even if water specimens were collected on the same day, there will be dierences
in storage me, distribuon me, water use paerns, and other factors.
•Two independently distributed random variables y
1
and y
2
have, respecvely, mean
values η
1
and η
2
and variances and
6.2. INDEPENDENT T-TEST FOR ASSESSING THE DIFFERENCE OF TWO AVERAGES
•The usual statement of the problem is in terms of tesng the null hypothesis that
the dierence in the means is zero:
•but we prefer viewing the problem in terms of the condence interval of the
dierence
lOMoARcPSD| 45903860
10
6.2. INDEPENDENT T-TEST FOR ASSESSING THE DIFFERENCE OF TWO AVERAGES
Usually the variances
esmator of variance:
lOMoARcPSD| 45903860
test stasc used
Degree of freedom
for t distribuon: for
the case where ଵ
and ଶ are
unknown:
Interval esmate of the dierence between two populaon means:
Margin of error
6.2. INDEPENDENT T-TEST FOR ASSESSING THE DIFFERENCE OF TWO AVERAGES
These can be pooled if they are of equal magnitude.
Assuming this to be true, the pooled esmate of the variance is:
lOMoARcPSD| 45903860
12
This is the weighted average of the variances,
where the weights are the degrees of
freedom of each variance.
To construct the (1−α)100% percent condence interval use the t stasc for α/2 and
ν=n
1
+n
2
−2 degrees of freedom
lOMoARcPSD| 45903860
6.3. HYPOTHESIS TESTS ABOUT THE DIFFERENCE BETWEEN THE PROPORTIONS OF
TWO POPULATIONS
•Similar applicaon for the proporon:
lOMoARcPSD| 45903860
14
lOMoARcPSD| 45903860
6.3. HYPOTHESIS TESTS ABOUT THE DIFFERENCE BETWEEN THE PROPORTIONS OF
TWO POPULATIONS
•This test stasc applies to large sample situaons where n
1
p
1
, n
1
(1 − p
1
), n
2
p
2
,
and n
2
(1 − p
2
) are all greater than or equal to 5.
•Interval esmate of the dierence between two populaon proporons:
lOMoARcPSD| 45903860
End of le 2.
Any quesons?
15
lOMoARcPSD| 45903860
EXERCISES
lOMoARcPSD| 45903860
tu@hcmiu.edu.vn 16
EXERCISES
lOMoARcPSD| 45903860
tu@hcmiu.edu.vn 17
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lOMoAR cPSD| 45903860 APPLIED STATISTICS COURSE CODE: ENEE1006IU Lecture 10: Chapter 6: t-Test
(3 credits: 2 is for lecture, 1 is for lab-work) 1 lOMoAR cPSD| 45903860 T-TEST
• For example, two methods for making a chemical analysis are compared to see if the
new one is equivalent to the older standard method; algae are grown under different
conditions to study a factor that is thought to stimulate growth; etc.
“Do two different methods of doing A give different results?”
“Can we be highly confident that the difference is positive or negative?”
“How large might the difference be?”
• One experimental design is to make a series of tests using treatment A and then to
independently make a series of tests using method B. Independent t-test
• A second way of designing the experiment is to pair the samples according to time,
technician, batch of material, or other factors that might contribute to a difference
between the two measurements Paired t-test lOMoAR cPSD| 45903860
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES
•Two samples are said to be paired when each data point in the first sample is
matched and related to a unique data point in the second sample.
•Paired experiments are used when it is difficult to control all factors that might influence the outcome.
•If these factors cannot be controlled, the experiment is arranged so they are
equally likely to influence both of the paired observations. Paired data are
evaluated using the paired t-test •The classical null hypothesis is:
“The difference between the two methods is zero.”
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES
•The paired t-test examines the average of the differences between paired observations 3 lOMoAR cPSD| 45903860
•Let (X11, X21), (X12, X22), … , (X1n, X2n) be a set of n paired observations where: -µ1 and
are the mean and variance of the population represented by X1 -µ2 and
are the mean and variance of the population represented by X2
Define the differences between each pair of observations as Dj=X1j - X2j, (j= 1, 2, …, n)
The Dj’s are assumed to be normally distributed with mean µD and variance
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES lOMoAR cPSD| 45903860
: the sample average of the n differences D1, D2, … , Dn SD: the
sample standard deviation of these differences : : : 5 lOMoAR cPSD| 45903860
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES Ha: Ha: Ha: lOMoAR cPSD| 45903860
6.1. PAIRED T-TEST FOR ASSESSING THE AVERAGE OF DIFFERENCES 7 lOMoAR cPSD| 45903860 End of file 1. Any questions? 8
6.2. INDEPENDENT T-TEST FOR ASSESSING THE DIFFERENCE OF TWO AVERAGES
•A t-test on the difference of the averages would conclude that A and B are not different.
•Sometimes it is not possible to pair the tests, and then the averages of the two
treatments must be compared using the independent t-test. Example: lOMoAR cPSD| 45903860
•Even if water specimens were collected on the same day, there will be differences
in storage time, distribution time, water use patterns, and other factors.
•Two independently distributed random variables y1 and y2 have, respectively, mean
values η1 and η2 and variances and
6.2. INDEPENDENT T-TEST FOR ASSESSING THE DIFFERENCE OF TWO AVERAGES
•The usual statement of the problem is in terms of testing the null hypothesis that
the difference in the means is zero:
•but we prefer viewing the problem in terms of the confidence interval of the difference lOMoAR cPSD| 45903860
6.2. INDEPENDENT T-TEST FOR ASSESSING THE DIFFERENCE OF TWO AVERAGES Usually the variances estimator of variance: 10 lOMoAR cPSD| 45903860 test statistic used Degree of freedom for t distribution: for the case where ଵ and ଶ are unknown:
Interval estimate of the difference between two population means: Margin of error
6.2. INDEPENDENT T-TEST FOR ASSESSING THE DIFFERENCE OF TWO AVERAGES
These can be pooled if they are of equal magnitude.
Assuming this to be true, the pooled estimate of the variance is: lOMoAR cPSD| 45903860
This is the weighted average of the variances,
where the weights are the degrees of freedom of each variance.
To construct the (1−α)100% percent confidence interval use the t statistic for α/2 and
ν=n1+n2−2 degrees of freedom 12 lOMoAR cPSD| 45903860
6.3. HYPOTHESIS TESTS ABOUT THE DIFFERENCE BETWEEN THE PROPORTIONS OF TWO POPULATIONS
•Similar application for the proportion: lOMoAR cPSD| 45903860 14 lOMoAR cPSD| 45903860
6.3. HYPOTHESIS TESTS ABOUT THE DIFFERENCE BETWEEN THE PROPORTIONS OF TWO POPULATIONS
•This test statistic applies to large sample situations where n1p1, n1(1 − p1), n2p2,
and n2(1 − p2) are all greater than or equal to 5.
•Interval estimate of the difference between two population proportions: lOMoAR cPSD| 45903860 End of file 2. Any questions? 15 lOMoAR cPSD| 45903860 EXERCISES lOMoAR cPSD| 45903860 tttu@hcmiu.edu.vn 16 EXERCISES lOMoAR cPSD| 45903860 tttu@hcmiu.edu.vn 17