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как первого S- состояния, так и первого Т-состояния, что обусловлено взаимодействием резонансно возбужденных локальных поверхностных плазмонов в наночастицах серебра с электронными состояниями молекул родамина 6Ж. Понижение температуры пленок ПВС с родамином 6Ж до 80 К позволило выявить релаксационные процессы в полимере, связанные с либрационными колебаниями мономерных звеньев в ПВС, соответствующие области р-релаксации. При этом наличие наночастиц практически не сказывается на процессах p-релаксации в полимере и, следовательно, ускорении или замедлении безызлучате-льной дезактивации возбужденных состояний.
Важным результатом данной работы также является обнаружение немонотонного характера температурной зависимости фосфоресценции, включающий интервалы как уменьшения, так и увеличения времени жизни T-состояний молекул родамина 6ж в пленке ПВС.
Результаты получены в рамках государственного задания Минобрнауки России № 3.809.2014/K.
Список литературы
1. Cao Y. C., Jin R., Mirkin C. A. Nanoparticles with raman spectroscopic fingerprints for DNA and RNA detection // Science. 2002. № 297. — P. 1536.
2. Burda C., Chen X., Narayanan R., El-Sayed M. The chemistry and properties of nanocrystals of different shape // Chem. Rev. 2005. № 105. — P.1025.
3. Климов В. В. Наноплазмоника. М., 2010. С.—480
4. Bryukhanov V.V. Effect of silver nanoparticles on singlet-singlet energy transfer dinamics of
luminofophores in thin films of polyvinyl alcohol / V.V. Bryukhanov, A.V. Tcibulnikova, I.G. Samusev, V.A. Slezhkin // J. Appl. Spectroscop. — 2014. — V. 81. — № 4. — P. 570-576.
5. Брюханов В.В. Взаимодействие поверхностных плазмонов наночастиц серебра на силохроме и шероховатых пленках серебра с электронно-возбужденными адсорбатами молекул родамина 6Ж / Брюханов В.В., Тихомирова Н.С., Горлов Р.В., Слежкин
B.А.// Известия Калининградского государственного технического университета. - 2011. - №23. -
C. 11-17.
6. Лакович Д. Основы флуоресцентной спектроскопии, М.: Мир, 1986, — С. 76
7. Bryukhanov V.V. Plasmonic enhancement and quenching of fluorescence and phosphorescence of anionic and cationic dyes in various environments / V.V. Bryukhanov, B.F. Minaev, A.V. Tcibulnikova, N.S. Tikhomirova, V.A. Slezhkin // J. Optical Techn. 2014. — Т. 81. — № 11. — С.7-14.
8. Слуцкер А.И., Поликарпов Ю. И., Васильева К.В.К определению энергии активации релаксационных переходов. ЖТФ. —2002. —Т.72. — №7. — С. 8691.
9. Минаев Б.Ф. Электронные механизмы активации молекулярного кислорода // Успехи химии. 2007. —Т. 76. —№ 11. — С. 1059-1083.
10. Рейтлингер С.А. Проницаемость полимерных материалов, М.: Химия, 1974. — С. 272
STATISTICS OF THE SCIENCES
Litvin Dmitry Borisovich
Candidate of Technical Sciences, Docent of Department of Mathematics, Stavropol Sami Atiyah SayyidAl-Farttoosi
PhD student College of Basic Education, Misan University, Iraq
Students are often intimidated by statistics. This brief overview is intended to place statistics in context and to provide a reference sheet for those who are trying to interpret statistics that they read. It does not attempt to show or to explain the mathematics involved. Although it is helpful if those who use statistics understand the math, the computer age has rendered that understanding unnecessary for many purposes. Practically speaking, students often simply want to know whether a particular result is significant, i.e. how likely it is that the obtained result may be attributable to something other than chance. Computer programs can easily produce numbers that allow such conclusions, if the student knows which tests to use and has an understanding of what the numbers mean. This summary is intended to help achieve that understanding [1].
As with all t tests, the one-sample t test assumes that the data be reasonably normally distributed, especially with respect to skewness. Extreme or outlying values should be carefully checked.
Before proceeding with the one-sample t test, we must verify the assumption of normality distributed data, by getting a histogram or a stemplot or a boxplot graph or by using normality test ( One Sample t test), see the results below The One-Sample T Test procedure: • Tests the difference between a sample mean and a known or hypothesized value
• Allows you to specify the level of confidence for the difference
• Produces a table of descriptive statistics for each test variable
Example: This example uses the file score.save. Use One Sample T Test to determine whether or not the mean score of math for the sample significantly differ from 75.
Note: The data used in one sample t test is a quantitative
data.
Before proceeding with the one-sample t test, we must verify the assumption of normality distributed data, by getting a histogram or a stemplot or a poxplot graph or by using normality test (One Sample t test), see the results below which shows that the distribution of the math score is a normal distribution. Histogram plot for math [2]. Sample Mean
The sample mean (X) is defined as the mean or average of a limited number of samples drawn from a population of experimental data [4]. The mean can be calculated manually or with the aid of a statistical function on a scientific calculator. The latter method is the most desirable and time efficient. Despite the use of wonderful technology, it is important to understand how the value is derived.
X = ( E Xk ) / n Where, xk is defined as the value of an individual experimental value, E Xk is the sum of all the experimental
values and n, is the number of experimental values used to obtain the sum.
For example, a certain experiment yielded the following data values for lead: 10 ppm, 8 ppm, 7 ppm, 11 ppm and 16 ppm. The mean value is calculated by the following:
(10 + 8 + 7 + 11 + 16) ppm/5 = 10.4 ppm = 10 ppm (use the appropriate significant digits)
15-
5-
0_l- I --jJ-- , ----1—г
50 60 70 80 90
math
control. Typically, a student t-test is used to indicate the difference between two means.
Case 1: If an accepted value, such as a Certified Reference Material (CRM), is known
This type of situation is used to compare an experimental mean with a value that is obtained from a sample, where the value is certified through analytical means known as a Certified Reference Material (CRM). CRMs are put through rigorous testing procedures to validate accurate concentrations levels and therefore there is a high degree of confidence is these analytically determined concentrations. In order to compare an experimental value with a CRM value to validate a method/procedure, the following t-test is utilized: ^ = x ± ts /sqrt(N)
If the equation is rearranged for the value of t: ± t = (x - ^)sqrt(N)/s,
where ^ is the value of the certified reference material, t is the student's t-value, obtained for N-1 degrees of freedom, at a pre-selected confidence interval, typically a 95% confidence interval. The t-values are obtained from a table similar to the one below:
Table 1
Values ^ for t at N-1 Degrees of Freedom for Various Confidence Intervals (CI)
N-1 90% CI 95% CI 99% CI
1 6.314 12.706 127.32
2 2.920 4.303 14.089
3 2.353 3.182 7.453
4 2.132 2.776 5.598
5 2.015 2.571 4.773
6 1.943 2.447 4.317
7 1.895 2.365 4.029
8 1.860 2.306 3.832
9 1.833 2.262 3.690
10 1.812 2.228 3.581
да 1.645 1.960 2.807
Standard Deviation
The term standard deviation (s) is used as a measure of precision. Precision describes how two or more numbers are in agreement if the exact same method or procedure is used. The standard deviation can be easily calculated using the statistical function on any calculator. But, again, understanding the mathematical derivation is important. Standard deviation is calculated by [5]:
? - i M_
Using the example from above, the standard deviation
is:
sqrt[(10-10) 2+(10-8) 2+(10-7) 2+(11-10) 2+(16-10) 2/5-1] = =sqrt[0+4+9+1+36/4] = 3.5 = 4 ppm
The mean and standard deviation for the experiment can be expressed as: (10 ± 4) ppm. Types of Student t-tests
A variety of student t-tests can be utilized to evaluate methods for purposes of method development or quality
Using the same data set utilized earlier for lead: 10 ppm, 8 ppm, 7 ppm, 11 ppm and 16 ppm. Assume there is a CRM value of 9.43 ppm for lead in a sandy soil sample. Case 1 can be used to compare whether or not the data for the given experimental method is considered reliable and valid in contrast to the CRM value: ± t = (x - ^)sqrt(N)/s
Plugging in the values: ± t = (10 ppm - 9.43 ppm)sqrt(5)/4 ppm ± t = 0.32
Consulting the t-table at the 95% confidence interval, at N-1, the t-value is 2.776. If the calculated t-value is lower than the tabulated t value at the 95% CI, there is not statistical difference. If the calculated t-value is higher than the tabulated t value at the 95% CI, there is a statistical difference. In this case, the calculated t-value is lower than the tabulated t-value and therefore the method is considered a valid procedure. Case 2: When the accepted value is unknown
When the accepted value is unknown, a paired t-test is used to determine the validity of the experimental number. Usually, a second mean is achieved using a different instrument, another laboratory or a secondary method within the same laboratory. The experiment t-value is calculated by:
± t = ((xi - X2)/sp)(NiN2/Ni + N2) *
where XX i is the mean of one data set, XX2 is the mean from the second data set and sp is called the pooled standard deviation given by:
sp = (si2(Ni-1) + s22(N2-1) + ... Sk2(Nk-1)/NT-k)*
Where the value of k is the number of experimental means used for comparison. For example, if there are two sets of experimental means, then the value of k is 2. Example:
Table 2
Lead Concentrations For Two Different Method Determinations Using ICP-MS From Lab A and Lab B
Lab A Data/ppm of Pb Lab B Data/ppm of Pb
17.1 17.2
16.2 17.1
14.6 17.0
22.8 19.0
18.7 18.3
X1 = 17.9 XX 2 = 17.7
S1 = 3.2 S2 = 0.9
sp = (39.7 + 3.0)/(I0-2) * = 2.3 ± t = (17.9 - I7.7)/2.3(5 x 5/5 + 5) * = (0.09)(1.6) = 0.1 The t-value from the table at a 95% confidence interval for 10 samples at N-I is 2.262. Since the calculated t-value is less than the tabulated t-value at a 95% confidence interval, there is no statistical difference between the two methods. Therefore, both methods are valid procedures. Rejection of Data Points
Often in research there are data points that seem out of range or questionable compared to the entire data set. It may be desirable to omit questionable data points from overall calculations. Questionable data that is omitted is called an outlier. However, omission of data points must be rigorously questioned using a statistical method called a Q-test. To
conduct the statistical test, the value of Q is compared to its nearest data point called a. A second variable called w, is the difference between Q and its furthest data point. A Q-test is determined by the following: Q = a/w
Considering, the original data set for lead: 10 ppm, 8 ppm, 7 ppm, ii ppm and i6 ppm, we may consider i6 ppm as a potential outlier. To test the validity of this assumption, the Q-test will be utilized:
Q = a/w = I6-II/I6-7 = 5/9 = 0.55 To assess the value of 0.55, one needs to refer to a table of rejection quotient for various confidence levels, similar to the one below:
Table 3
Rejection Quotients (Q) at Various Confidence Intervals
# of Observations Q90 Q95 Q99
3 0.941 0.970 0.994
4 0.765 0.829 0.926
5 0.642 0.710 0.821
6 0.560 0.625 0.740
7 0.507 0.568 0.680
8 0.468 0.526 0.634
9 0.437 0.493 0.598
10 0.412 0.466 0.568
As there are five data points with no loss of degrees of freedom, n = 5 then Q = 0.710 at 95% CI. If a calculated Q-value is greater than the tabulated Q-value, the outlier can be rejected. However, if a calculated Q-value is less than the tabulated Q-value, then the outlier cannot be rejected as it is considered a valid data point.
Referring back to the example, the calculated Q-value = 0.55, the tabulated Q-value = 0.710 at 95% CI. Therefore, the calculated Q value < tabulated Q value, and, the value of i6 ppm cannot be rejected.
F-test: Comparison of Precision Measurement An F-test is a simple calculation to compare the precision of two sets of measurement. The sets do not have to
be obtained from the identical sample, so long as both samples are sufficiently similar that any indeterminate errors can be considered the same. An F-test can provide insights into two main areas: 1) Is method A more precise than method B? 2) Is there a difference in the precision of the two methods? To calculate an F-test, the standard deviation of the method which is assumed to be more precise is placed in the denominator, while the standard deviation of the method which is assumed to be least precise is placed in the numerator.
Using the two-piece data set for lead obtained above, the standard deviations of si = 3.2 ppm (least precise) and s2 = 0.9 ppm (more precise) were obtained.
F= si2/ s22 = (3.2)2/(0.9)2 = 10.2/0.8 = 12.8 To further analyse this resultant F-test value, reference to a table of critical values for F is essential. A similar table is found below:
Table 4
Critical Values For F At A 5% Level
Degrees of Freedom (Denominator) Degrees of Freedom (Numerator)
2 3 4 5
2 T9.00 T9.T6 T9.25 T9.30
3 9.55 9.28 9.T2 9.0T
4 6.94 6.59 6.39 6.26
5 5.79 5.4T 5.T9 5.05
Each data set had five degrees of freedom and hence the tabulated F-value is 5.05. In comparison to the calculated F-test, the calculated value of 12.8 is greater than the tabulated value of 5.05. Therefore, it is demonstrated that the more precise method is indeed derived from data set number two [3].
List of references
1. www.csub.edu/~bhartsell/StatisticsReview.doc
2. site.iugaza.edu.ps/nbarakat/files/2010/02/part4.doc
3. Statistical tables where derived from: Douglas, A.S; West, D.M.; Holler F.J., 1992, Fundamentals of Analytical Chemistry, Sixth Edition. Saunders College Publishing, Florida, USA.
4. Гулай Т.А., Долгополова А.Ф., Литвин Д.Б. Анализ и оценка приоритетности разделов математических дисциплин, изучаемых студентами экономических специальностей аграрных вузов.//Вестник АПК Ставрополья. 2013. № 1.
5. Гулай Т. А., Долгополова А. Ф., Литвин Д. Б. Совершенствование профессиональной подготовки экономистов через направленность содержания математического образования // Аграрная наука, творчество, рост: сб. тр. Междунар. науч.-практ. конф. (Ставрополь, 08-14 февраля 2013 г.) / СтГАУ. Ставрополь, 2013. Т. 2. С. 252-254.
ANOVA F TESTS FOR WITH P TREATMENTS AND B
Litvin Dmitry Borisovich
Candidate of Technical Sciences, Docent of Department of Mathematics, Stavropol Sabah Hasan Jasim Al-saedi
PhD student College of Basic Education, Misan University, Iraq
The t test is commonly used to test the equality of two population means when the data are composed of two random samples. We wish to extend this procedure so that the equality
of r > 2 population means can be tested using r independent samples. Thus the hypothesis and the alternative are
H0: M = M2 = ... = Mr
H1 : at least two means are not equal,
where
Mj, j = 1,2,...,r is the mean of the j-th population.
classification is sometimes referred to as a completely randomized design [5].
Samples from each of the r populations are collected.
Xij = the i-th observation receiving treatment j
n,
i = 1,2,..., j; j = 1,2,..., r _ 1 nj
Xa. = —V X . = mean of sample j
It is not hard to imagine situations in which it is of interest to compare a number of means. For example, 5 varieties of corn are available, and it is to be determined whether or not the average yield from each variety is the same; a company is testing 3 brands of bicycle tires and wants to know if the average life of each brand is the same; 4 teaching methods are being investigated for their effectiveness; an automotive company wants to determine which of 4 seat-belt designs would provide the best protection in the event of a head-on collision; a drug company would like to compare the effectiveness of 6 different drugs for treating diabetes.
In designing an experiment for a one-way classification, units are assigned at random to any one of the r treatments under investigation. For this reason, the one-way
^2 =
nj -
(XtJ - XQj )2 = variance of sample j
1 7=1
1 Г nj Г X»=17 E 2X, N=£n j
ly 3=1 /=1 3=1
1 r nJ
s 2 =-V V (X.., - X. )2 = variance of all N observations
N -
TE E(Xj - X)2 -
1 /=1 /=1
M, andCT , j = 1,2,...,r j j , denote the mean and
variance of population j.
Here's one way the data can be arranged once it is
collected
n
