HANOI UNIVERSITY FACULTY OF MANAGEMENT AND TOURISM -o0o- STATISTICS FOR ECONOMICS Is there any difference in the number of students per teacher over years? Tutor: Ms Lê Thị Ngọc Tú Tutorial: – AC 09 Tutorial time: Tuesday – 12.30 – 14.00 Group members: Nguyễn Huyền Trang Trần Thu Hằng Nguyễn Thị Tươi Lê Thị Minh Thành 0904010116 0904010030 0904010112 0904010098 Nguyễn Thị Huệ Nguyễn Thị Hồng Nga Trần Thi Thanh Vân Nguyễn Thị Mai 0904010043 0904010077 0604040183 0904010065 TABLE OF CONTENTS Scenario I.Methodology 1.Data collection 2.Approach II Analysis and discussion .4 3.Check the required condition 1.1.Normality 1.2.Variances equality 4.Hypothesis testing 2.1 Testing block means .6 2.2 Testing treatment means 5.Discussion of finding III.Limitation .9 IV.Recommendation and conclusion .9 6.Recommendation 7.Conclusion 10 Reference i Appendixes ii A.Calculating the sample variance ii B.Check the variances equality iii C.ANOVA (using Excel) iv D Histograms v E Data from GSO vii Scenario In recent years, along with an increasing demand in human resources, a growing number of universities have plan to open new faculties as well as increase the number of Case study - ANOVA student admissions for these hot sectors However, it is undeniable that the mismatch between the number of students’ enrollment and teachers/lecturers’ quantity has large effect on the quality of education and training To be aware of this important issue, our group decided to find out whether there are any differences in the number of students per teacher from 2005 to 2009 (particularly 2005, 2007 and 2009) by using statistical technique (2-way ANOVA) The available data is blocked into six main regions in Vietnam After conducting the test, the result show that during this 6-year period, despite the changes in both number of students and teachers, the number of students per teacher is nearly the same, which lead to our conclusion that there is no difference among three years I Methodology Data collection As the problem objective is to test whether there are changes in the amount of students per teacher in recent years in Viet Nam, to be more detail we conduct the test over three years including 2005, 2007, and 2009 Moreover, the data type is quantitative; Case study - ANOVA we decided to use the analysis of variance The data was collected from the Vietnam General Statistics Office website (shown in Appendix E) However, we pointed out that many other factors may affect to the result of our test As a result, the variability within the samples might be large In order to reduce the variation in each year, we made the survey according to blocks and then did the test Therefore, we took a random sample of six regions containing Red River delta, Northern midlands & mountainous, Northern Central and Central Coastal, Highlands, South East, and Mekong River delta to test the changes in the rate of student over one teacher in those areas over three years Nevertheless, because it was so difficult to conduct the experiment on those areas, we continued using excel to select randomly one province in each area to be on behalf of that region And thereafter, we got the result of six provinces: Hai Phong, Son La, Da Nang, Kon Tum, Dong Nai, and the last one is Kien Giang Thus, there are six blocks containing six regions and three treatments are three years in this test The experimental design used here is a randomized block design, which treatments are the three years 2005, 2007, 2009 After doing the test, the following table was produced: 2005 2007 2009 Red River delta 23.04452467 28.10416667 28.43558606 Northern midlands & mountainous 21.81818182 31.32592593 10.34782609 North Central and Central Coast 45.16666667 33.19047619 27.26348748 Highlands 24.68253968 12.05464481 38.86703383 South East 19.89583333 25.53491436 37.99269006 Mekong River delta 14.95890411 8.356495468 11.10789474 Approach In order to indicate whether differences exist among the number of students over the quantity of teachers over three years, it is necessary to check the required conditions for using F-test of two-way ANOVA, which are the random variable is normally distributes and the population variances are equal We will check each condition one by one Case study - ANOVA II Analysis and discussion Check the required condition 1.1 Normality As you can see from the histogram in Appendix D, the three populations are non normal, in order to use way ANOVA, we assume that all of them are normally distributed 1.2 Variances equality Since the best estimator of population variance is the sample variance, we applied the F - test to compare the variability of two populations (biggest versus smallest ones, shown in Appendix B) With α = 5%, the F-values of the three tests are higher than 0.05 Therefore, it can be inferred that the variances are equal For its applicability, two-way ANOVA is a procedure that testes to determine whether differences exist among two or more population means It enables to measure how much variation is attributable to difference among populations and how much variation is attributable to differences within populations By designing a randomized block design experiment, it reduces the within treatment variation so as to more easily detect difference among the treatment means However, the technique only allows testing for a difference rather than indicating which population means exceed others Case study - ANOVA After calculating the variance (shown in appendix A), the largest variance is that one in 2009 while the smallest one is in 2007, so we use F-test to make inference about those two population variances Testing hypothesis: HO : σ 12 =1 σ 22 σ 12 HA : ≠1 σ2 Test statistic: s12 F= s2 v is F-distributed with = n1 − and v2 = n2 − Significance level: α = 0.05 Decision rule: Reject Ho if F > Fα/2, v1, v2 = F.025, 2, = 39 or F < F1-α/2, v1, v2 = 1/F.025, 2, = 0.0256 Value of test statistic: As shown in Appendix B: F = 0.69 Conclusion: Since 0.0256 < F = 0.69 < 39, not reject Ho Therefore, there is not enough evidence to conclude that the population variances differ Case study - ANOVA Hypothesis testing 2.1 Testing block means Testing hypothesis: Ho: Block means are all equal Ha: At least tow block means differ Test statistic: F= MSB MSE is F-distributed with ν1 = b – and ν2 = n – k – b + Significance level: α = 0.05 Decision rule: Reject Ho if F > Fα, b -1, n – k – b +1 = F.05, 5, 10 = 3.33 Value of test statistic: As shown in the ANOVA table (Appendix C) F = 1.98995 Conclusion: Since F = 1.98995 < 3.33, not reject Ho Therefore, there is not enough evidence to conclude that block means differ, which indicate that we can use blocks to remove the variability and two-way ANOVA can be conducted Case study - ANOVA 2.2 Testing treatment means Testing hypothesis: H O : µ1 = µ = µ3 H A : At least treatment means differ Test statistic: F= MST MSE is F-distributed with ν1 = k – and ν2 = n – k – b +1 Significance level: α = 0.05 Decision rule: Reject Ho if F > Fα, k – 1, n – k – b + = F.05, 2, 10 = 4.10 f(F) Rejection Region 0.11241 4.10 Value of test statistic: As shown in the ANOVA table (Appendix C): F = 0.11241 Conclusion: Since F = 0.11241 < 4.10, we not reject Ho Hence, there is not sufficient evidence to conclude that differences exist among the three years Case study - ANOVA Discussion of finding It is obvious from the hypothesis tests that there is not enough evidence to reject the null hypothesis, which assumes that there is no difference between the ratios of students/teacher in Vietnam over five year period From the result extracted from the data analysis section, there is also no difference among the block means representing the population of six main regions in Vietnam Therefore, it is quite easy to recognize the balance state through these six areas If the test were not conducted, people may think that the ratio of students per teacher increases over the years because of the student growth in Vietnam The fact shows that due to high demand in high quality human resource to meet challenges of economic growth, many universities/colleges have increased the number of admission year by year To be aware of that fact, education units have had plan to recruit more teachers to keep up with the increase in number of students and remain/improve teaching quality This fact somehow explains the reason for unchanged number of students per teacher over the years However, compared with the world’s standard (15-20 students/teacher) and the goal of Ministry of Education and Training (20 students/teacher), the current ratio in Vietnam is still much higher with 28 students/teacher Therefore, we need to increase the number of teachers to improve the quality of our country‘s education Besides, teachers’ quality (degree, teaching skills, etc) which directly affects education quality should be concerned about From the result extracted from the data analysis section, there is also no difference among the block means representing the population of six main regions in Vietnam So, it is quite easy to recognize the balance state through these six areas Case study - ANOVA III Limitation Although we tried to test with our best effort, some limitations still happened These following limitations can reduce our test’s accuracy: • Lack of information: it is difficult to find information through out longer periods (5-year periods in stead of 1-year periods as we showed previously).The 1-year periods can be too short time so that this limitation can reflect inaccuracy in changing the number of professors As a result, our conclusions may be not much exactly • Rejection regions: we chose α = 0.05, which might lead to type II error However, we believed that it is not affecting our result so much • Time consuming: because checking consumptions are necessary for testing so we spend lots of time to check the normality of populations and the equality of its variance Fortunately, histograms drawn resulting normally distributed populations as we expected Moreover, we also check SSB to ensure that there is no difference between blocks • Normality: In order to follow the way ANOVA test above, we have assumed that the three populations are normally distributed IV Recommendation and conclusion Recommendation In the recent years, the number of student increases continuously in universities As we expected, the ratio of the professors and their students does not change from year Case study - ANOVA to next year, which means that it does not have strong influences on the quality of teaching and studying However, we still have some recommendation in order to improve those qualities + Reinforcing high qualified professors: since the number of students increases in universities, it creates a lot of pressure on education The lack of high qualified teachers is inevasible Therefore, reinforcing high qualified professors are the first principles + Motivating teachers: the teachers should be facilitated studies with suitable compensations Beside, creating good relationships between teachers and their students are respected also Thus, that reduce a large number of teachers quit their jobs + Changing from traditional classes to new model ones: let Hanoi University be an example, the students and teachers attend at five lectures and five tutorials each week Consequently, the professors and their students have extra time for self-study + Flexible time: both teachers and student as well can involve in the social activities, voluntary event, and part-time jobs in order to gain practical experiences, soft skills like communication skills In addition, universities can provide enough facilities and equipments for teaching Conclusion In conclusion, the report carried out on the purpose of dealing with a statistics question: whether there exist any differences in the number of students per teacher through 5-years period of time from 2005 to 2009 in six main regions in Vietnam including Red River delta, Northern midlands & mountainous, Northern Central and Central Coastal, Highlands, South East, and Mekong River delta The findings drawn from this study shows that there are not differences from the number of students per over 10 Case study - ANOVA year in regions which we indicate above It also means that Vietnamese university education can provide enough teachers to meet the need of social in general and the increase in enrolment target through years However, we still need some recommendation in order to improve the education system as shown in our report During the time we were conducting the research, some limitation occurred which lead to inaccuracy result In addition, because of the characteristic of ANOVA test and time consuming, we can not show the whole picture of the issue for example, the trend of enrolment target, change in method and model class, etc If by any chance our report has aroused interest in other researchers about the same topic, we hope that future studies would be conducted on a larger time scale, with more detailed data, and with further knowledge of statistic 11 Case study - ANOVA Reference • General Statistic Office, Number of teachers, students in universities and colleges by province, http://www.gso.gov.vn/default_en.aspx?tabid=474&idmid=3&ItemID=10207 • http://vietbao.vn/Tuyen-sinh/Chi-tieu-tuyen-sinh-vao-cac-truong-DH-CD-nam2005/30050060/290/ • http://vietbao.vn/Giao-duc/Chi-tieu-tuyen-sinh-cac-truong-DH-nam-2007/70079320/202/ i Case study - ANOVA Appendixes A Calculating the sample variance SUMMARY Count Red river delta Sum 79.5843 Northern midlands and mountains areas 63.4919 21.164 110.341 Northern Central area and Central coastal area 105.621 35.2069 83.1804 Central highlands 75.6042 25.2014 179.928 South East 83.4234 27.8078 85.7486 Mekong river delta 34.4233 11.4744 10.9987 2005 2007 6 149.567 138.567 24.9278 23.0944 109.518 107.965 2009 154.015 25.6691 156.604 ii Average 26.5281 Variance 9.12889 Case study - ANOVA B Check the variances equality Mean Variance Observations df F P(F