COMPARISON OF THE WAYS OF ACCEPTANCE STUDENTS AT UNIVERSITY

The paper reports on an analysis of the entrance examinations at the Faculty of Informatics and Statistics at University of Economics in Prague. Applicants can be accepted to this faculty by three ways. The aim of this paper is to compare these ways of acceptance students at the Faculty of Informatics and Statistics at University of Economics in Prague and to study dependence of the results of entrance examinations in mathematics on test variants. Results of this analysis can be used for improvement of the entrance examinations at the Faculty of Informatics and Statistics at University of Economics in coming years.


Introduction
Applicants can be accepted to study Faculty of Informatics and Statistics at University of Economics in Prague by three ways: • on the base of tests in mathematics and English, which is used at University of Economics in Prague (VSE tests) • on the base of the national comparative exams -the tests of general academic prerequisites (SCIO tests) • without entrance examinations (on the base of results in mathematics and English at grammar school etc.).
The relations between the ways of acceptance applicants to study Faculty of Informatics and Statistics and study results in mathematics are studied in this paper.Similar problem (the dependence of study results and results of the learning potential tests) is solved in Kubanová and Linda (2012), Linda and Kubanová (2013).On the other hand, analysis of the study results in basic courses in mathematics at University of Economics is in Kaspříková (2012), Otavová and Sýkorová (2014).This paper is an extended version of the paper Klůfa (2015) -other group of students was analyzed.
In the second place we shall study dependence of the results of entrance examinations in mathematics on test variants.The math tests are prepared by the Department of Mathematics of the Faculty of Informatics and Statistics.These tests are the multiple choice question tests.The tests in mathematics have 10 questions for 5 points and 5 questions for 10 points (100 points total).Questions are independent.Each question has 5 answers (one answer is correct), wrong answer is not penalized.
The number of points in the test in mathematics can be Jindřich Klůfa  2 and Table 4.
Some statistical methods are used for an analysis of the entrance examinations at the Faculty of Informatics and Statistics -see e.g.Anděl (1978), Rao (1973).
For study dependence of number of points in the test in mathematics on test variants we shall use 2 where r is number of rows, s is the number of columns in contingency and where 2 (( 1 hypothesis of independence is rejected at significance level, which is asymptotically equal to . α For comparison of the ways of acceptance applicants to study we shall use nonparametric Kruskal-Wallis test.When (statistic Q see e.g.Anděl (1978: 231)) where 2 ( 1) χ distribution for 1 k − degrees of freedom ( k is number of the ways of acceptance applicants), hypothesis "distribution of number of points in mathematics is the same for all three ways of acceptance students" is rejected at significance level, which is asymptotically equal to .α Further for comparison of the ways of acceptance applicants to study we shall use a one-way analysis of variance (ANOVA).We shall verify the validity of the null hypothesis: mean number of points in mathematics is the same for all three ways of acceptance students.When (the test statistic F see e.g.Rao  (1973: 280))  Some parameters of distribution of number of points in mathematics are in Table 3. From

α =
With high probability (approximately 0.99), we can say that there are significant differences between ways of acceptance students to study Faculty of Informatics and Statistics (VSE tests, SCIO tests, other).
Remark.Similar result gives ANOVA: null hypothesis "mean number of points in test in the course Mathematics for informatics and statistics is the same for all three ways of acceptance students" is rejected ( 7.088 4.822) at significance level 0.01 α = -see Tab. 5. Assumption of ANOVA: null hypothesis "variance of number of points is the same for all three ways of acceptance students" is not rejected at 1% significance level (Bartlett's test -see e.g.Anděl (1978: 155)).Since this hypothesis is rejected at 5% significance level, we used also corresponding nonparametric Kruskal-Wallis test.Finally we shall study which pairs of averages differ significantly.

Sum of Squares
We use Scheffe's method -see e.g.Anděl (1978: 154).Pairs of averages differ significantly if absolute value of difference in averages exceeds critical value (see Tab. 2 (n 1 =42, n 2 =11, n 3 =51) and Tab. 5) Printed ISSN: 2336-2375 From Tab. 6 it is seen that a significant difference is at 1% significant level only between SCIO tests and Other.
On the other hand, results in mathematics in the course Mathematics for economists (ident 4MM101) sorted according to the ways of acceptance applicants are in Table 4 (see Appendix).We shall test null hypothesis i.e. mean number of points in test in the course Mathematics for economists is the same for all three ways of acceptance students (VSE tests, SCIO tests, other).To verify the validity of the hypothesis (10) we use ANOVA.In the first step we verify assumption (the same variance of number of points in ways of acceptance) of this method by Bartlett's test.Test statistic B (see e.g.Anděl (1978: 155)) is χ distribution for 2 degrees of freedom and significance level Results of ANOVA we got with MS Excel -see Tab. 8.

Discussion
Similar problem as in this paper is solved in Kubanová and Linda (2012).There is studied the dependence of study results and results of the learning potential tests (SCIO tests).The insignificant correlation was detected between results in learning potential test and study results.In this paper was solved other problem -dependence between study results in mathematics and SCIO tests (ways of acceptance students) has been proven.
On the other hand, analysis of the study results in basic courses in mathematics at University of Economics is in Otavová and Sýkorová (2014).There is studied whether the score from final test depends on the score from mid-term test.Similar methods as in this paper show that dependence between the score from final test and the score from mid-term test exists.
The multiple choice question tests for entrance examinations at University of Economics in Prague from probability point of view are analysed in Klůfa and Kaspříková (2012).Disadvantages of such type of test is that a student can obtain certain number of points in the test purely by guessing the correct answers.Results of the paper show that risk of success of students with lower performance levels is negligible.The analysis of entrance examinations in this paper (the dependence on the test variants) also shows that these tests are suitable for entrance examinations at University of Economics.
Relation between results of the entrance examination test in mathematics and examination in mathematics at University of Pardubice is studied in Linda and Kubanová (2013).This paper had demonstrated the dependence of the test results in math on the results of entrance tests in mathematics (in contrast to SCIO tests).This finding should lead to the conclusion that the schools should focus on admission process, students should be accepted on the basis of own admission.Similar results we obtain also in this paper.

Conclusion
From 2 χ test of independence in contingency table it follows that the number of points in the test in mathematics does not depend on the test variant.From the results of this paper we can say that significant changes in test variants in mathematics in the coming years are not needed.
There are significant differences between students which were accepted to study Faculty of Informatics and Statistics on the base VSE tests, SCIO test and without entrance examinations from the point of view test score in mathematics (see also Figure 2).The best results from the point of view number of points in Printed ISSN: 2336-2375 test in mathematics in the course 4MM103 achieved by students which were accepted on the base of good results in mathematics and English at grammar school (without entrance examinations).
On the other hand, the differences between average number of points in test in the course Mathematics for economists (ident 4MM101) are not statistically significant.
For the eventual changes in the admission process, it would be useful to test the differences between ways of acceptance students to study Faculty of Informatics and Statistics in other courses.
From results of this paper it seems that applicants should be accepted to study Faculty of Informatics and Statistics at University of Economics in Prague on the basis of own admission process.

Fig. 1 :
Fig. 1: Distribution of number of points in test in mathematics in 2013 -test variants A6, A7, B2, B3 (histogram) Differences between the ways of acceptance applicants Now we shall compare the ways of acceptance applicants to study Faculty of Informatics and Statistics.Results in mathematics in the course Mathematics for informatics and statistics sorted according to the ways of acceptance applicants are in Table 2 (see Appendix).We shall test null hypothesis H o : distribution of number of points in mathematics is the same for all three ways of acceptance students (VSE tests, SCIO tests, other)

Fig. 2 :
Fig. 2: Average number of points in test in mathematics in the course 4MM103 Mathematics for economists (see last column of Tab.7) are not statistically significant).

Fig. 3 :
Fig. 3: Distribution of the final results (test + oral exam) in mathematics in course 4MM101 -ways of acceptance (histogram) 1* 1* Department of Mathematics, Faculty of Informatics and Statistics, University of Economics, W. Churchill Sq. 4, Prague, 130 67, Czech Republic, +420 224 094 244, klufa@vse.czPrinted ISSN: 2336-2375 summer semester of the 2013/2014 academic year in the course Mathematics for informatics and statistics (ident 4MM103) and the results of 472 students in winter semester of the 2014/2015 academic year in the course Mathematics for economists (ident 4MM101) are analysed.The number of points in the test in the course Mathematics for informatics and statistics and in the course Mathematics for economists can be in interval [0,40], the multiple choice question tests are not used.The analysed data are sorted according to ways of acceptance students to study Faculty of Informatics and Statistics at University of Economics in Prague -see Table

Dependence on test variants
hypothesis is rejected at significance level .Results of the entrance examinations in mathematics in 2013 are in Table 1 (for example 6) α Results

Distribution of number of points in test in mathematics (contingency table)
Marek (2013)sis H o is not rejected at approximately 5% Printed ISSN: 2336-2375 significance level.Moreover p value is 0.168 (null hypothesis H o is not rejected also at 16% significance level).For calculation we used MS Excel version 10 -seeMarek (2013).We can say that the number of points in the test does not depend on the test variant.

Descriptive statistics for number of points in the course 4MM103
Table 3 it seems that the distributions of number of points in mathematics (VSE tests, SCIO tests, other) differ, i.e. hypothesis H o is not valid.For objective decision we shall use a statistical test.To verify the validity of the hypothesis H o we shall use Kruskal-Wallis nonparametric test.Value of statistic Q is ( Q is calculated on the base of rank of values in Table 2 -see Anděl (1978: 231)) χ distribution for 2 degrees of freedom.Critical value of 2 χ distribution for two degrees of freedom and significance level