Is there any difference in the number of students per teacher over years?

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: 4 – 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................................................................................................1
I.Methodology......................................................................................2
1.Data collection..................................................................................................2
2.Approach...........................................................................................................3

II. Analysis and discussion.................................................................4
3.Check the required condition............................................................................4
1.1.Normality...................................................................................................4
1.2.Variances equality......................................................................................4
4.Hypothesis testing.............................................................................................6
2.1. Testing block means.................................................................................6
2.2. Testing treatment means...........................................................................7
5.Discussion of finding........................................................................................8

III.Limitation.......................................................................................9
IV.Recommendation and conclusion.................................................9
6.Recommendation..............................................................................................9
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
1. 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;

2

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

2. 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.

3


Case study - ANOVA
II.

Analysis and discussion
3. 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 2 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.

4


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
1. Testing hypothesis:

HO :

σ 12

=1
σ 22

σ 12
HA : 2 ≠1
σ2
2. Test statistic:

s12
F= 2
s2

v
is F-distributed with 1

= n1 − 1

and

v2 = n2 − 1

3. Significance level: α = 0.05
4. Decision rule:
Reject Ho if F > Fα/2, v1, v2 = F.025, 2, 2 = 39 or F < F1-α/2, v1, v2 = 1/F.025, 2, 2 = 0.0256
5. Value of test statistic:
As shown in Appendix B: F = 0.69
6. Conclusion:
Since 0.0256 < F = 0.69 < 39, not reject Ho.
Therefore, there is not enough evidence to conclude that the population variances differ.

5


Case study - ANOVA
4. Hypothesis testing
2.1. Testing block means
1. Testing hypothesis:
Ho: Block means are all equal
Ha: At least tow block means differ
2. Test statistic:

F=

MSB
MSE is F-distributed with ν1 = b – 1 and ν2 = n – k – b + 1

3. Significance level: α = 0.05
4. Decision rule:
Reject Ho if F > Fα, b -1, n – k – b +1 = F.05, 5, 10 = 3.33
5. Value of test statistic:
As shown in the ANOVA table (Appendix C) F = 1.98995
6. 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.

6

Case study - ANOVA
2.2. Testing treatment means
1. Testing hypothesis:

H O : µ1 = µ 2 = µ3
H A : At least 2 treatment means differ
2. Test statistic:
F=

MST
MSE

is F-distributed with ν1 = k – 1 and ν2 = n – k – b +1

3. Significance level:
α = 0.05
4. Decision rule:
Reject Ho if F

> Fα, k – 1, n – k – b + 1 = F.05, 2, 10 = 4.10

f(F)

Rejection
Region

0

0.11241

4.10

5. Value of test statistic:
As shown in the ANOVA table (Appendix C): F = 0.11241
6. Conclusion:
Since F = 0.11241 < 4.10, we do not reject Ho.
Hence, there is not sufficient evidence to conclude that differences exist among
the three years.

7


Case study - ANOVA
5. 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.

8


Case study - ANOVA
III.

Limitation

Although we tried to do 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 2 way ANOVA test above, we have assumed
that the three populations are normally distributed.

IV.

Recommendation and conclusion
6. 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

9


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.

7. 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, />
•

/>
•

/>
i


Case study - ANOVA

Appendixes
A. Calculating the sample variance
SUMMARY

Count

Red river delta

3

Sum
79.5843

Northern midlands and mountains areas

3

63.4919

21.164

110.341

Northern Central area and Central coastal area

3

105.621

35.2069

83.1804

Central highlands

3

75.6042

25.2014

179.928

South East

3

83.4234

27.8078

85.7486

Mekong river delta

3

34.4233

11.4744

10.9987

2005
2007

6
6

149.567
138.567

24.9278
23.0944

109.518
107.965

2009

6

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<=f) one-tail
F Critical one-tail

F-Test Two-Sample for Variances
Variable 1
23.09443724
107.9649341
6
5
0.6894119
0.346572174
0.1980069

Variable 2
25.66908638
156.6043958
6
5

Since 0.198 < F = 0.689 < 5, we do not reject Ho.
There is enough evidence to conclude that the two population variances are the same.

iii


Case study - ANOVA
C. ANOVA (using Excel)

Source of Variation

SS

df

MS

F

P-value

F crit

Rows

932.8621

5

186.572

1.98995

0.16583

3.32583

Columns

21.07898

2

10.5395

0.11241

0.89479

4.10282

Error

937.5724

10

93.7572

Total

1891.514

17

iv


Case study - ANOVA
D. Histograms
•

•

Population 1: 2005
Bin
10
20
30
40
50
More

Frequency
0
2
3
0
1

0

Bin
10
20
30
40
50
More

Frequency
1
1
2
2
0
0

Population 2: 2007

v


Case study - ANOVA
•

Population 3: 2009
Frequenc
Bin

y

10

0

20

2

30

2

40

2

50

0

More

0

vi


Case study - ANOVA

E. Data from GSO

Whole country
Red river delta

Num

1 Hà Nội
2 Hà Tây
3 Vĩnh Phúc
4 Bắc Ninh
5
6
7
8

Quảng Ninh
Hải Dương
Hải Phịng
Hưng n

9 Thái Bình
10 Hà Nam
11 Nam Định
12 Ninh Bình
Northern midlands and moutain areas
1 Hà Giang
2 Cao Bằng
3 Bắc Kạn
4 Tuyên Quang

2007
Teacher Student
192843
61321
6
25384
791671

S/T

16476

606207

36.793336

1404

29435

20.9651

536

17704

522

2008

Teacher Student
167570
60651
0
25310
695089
17065

529211

33.029851

568

18384

7624

14.605364

632

11676

896

8100

9.0401786

811

9272

761
1776
624

9677
49913
22875

12.716163
28.104167
36.658654

848
1862
907

13437
51070
22195

621

8409

13.541063

612

7222

118

3922

33.237288

268

3668

1517

27081

17.851681

1504

27590

133
4863
71

724
112385

2134

5.443609
30.056338

233
5702
65

1364
105105
1001

107

1410

13.17757

110

1734

212

2080

9.8113208

45

967

80

530

6.625

73

925

vii

S/T

31.01148
5
32.36619
7
18.47468
4
11.43279
9
15.845519
27.427497
24.470783
11.80065
4

13.68656
7
18.34441
5
5.8540773
15.4
15.76363
6
21.488889
12.67123
3

2009
Teacher Student

S/T

65115

1796174

26409

725976

18083

541671

29.954709

646

19576

30.303406

543

14530

26.758748

870

10277

11.812644

876
1894
963

13312
53857
24067

15.196347
28.435586
24.991693

613

8450

13.784666

315

4070

12.920635

1372

34802

25.365889

234
5978
71

1364
120033
1441

5.8290598

97

1571

16.195876

45

688

15.288889

73

905

12.39726

20.295775


Case study - ANOVA
5 Lào Cai
6 Yên Bái
7 Thái Nguyên
8 Lạng Sơn
9 Bắc Giang
10 Phú Thọ
11 Lai Châu
12 Sơn La
13 Hịa Bình

Northern Central area and Central
coastal area
1 Thanh Hóa
2 Nghệ An
3
4
5
6
7
8

Hà Tĩnh
Quảng Bình
Quảng Trị
Thừa Thiên-Huế
Đà Nẵng
Quảng Nam

9 Quảng Ngãi
10 Bình Định
11 Phú Yên

97

1917

19.762887

81

1552

70
2437

829
70666

11.842857
28.997128

109
2929

935
69822

148

1252

8.4594595

166

883

228

3592

15.754386

223

2333

725

10519

14.508966

1112

9959

124

2547

20.540323

187

2838

405

12687

31.325926

417

10226

159

2222

13.974843

185

1930

9601

316394

9640

268741

700

16646

23.78

808

15276

1282

41358

32.26053

1134

40293

162

1172

7.2345679

157

2555

138
78
1952
2394
650

4889
1272
97154
79458
3771

35.427536
16.307692
49.771516
33.190476
5.8015385

148
79
2009
2785
537

4952
1171
52141
82229
6984

403

5553

13.779156

280

5769

609

27751

45.568144

628

19825

329

4192

12.741641

241

4693

viii

19.16049
4
8.5779817

23.83817
5.319277
1
10.46188
3
8.9559353
15.17647
1
24.522782
10.43243
2
18.90594
1
35.53174
6
16.27388
5
33.459459
14.822785
25.953708
29.525673
13.005587
20.60357
1
31.56847
1
19.47302
9

81

714

8.8148148

111
3019

1264
75433

11.387387
24.986088

166

3188

19.204819

244

3001

12.29918

1031

13820

13.404462

214

2869

13.406542

23

238

10.347826

471

11706

24.853503

332

3195

10866

292413

26.910823

830

16022

19.303614

1325

39175

29.566038

167
148
80
2076
3135

2854
5039
1246
56599
90889

17.08982
34.047297
15.575
27.263487
28.991707

634

10616

16.744479

375

6270

16.72

696

22994

33.037356


Case study - ANOVA
12 Khánh Hịa
13 Ninh Thuận
14 Bình Thuận
Central highlands
1 Kon Tum
2 Gia Lai
3 Đắk Lắk
4 Đắk Nông
5 Lâm Đồng
South East

1 Bình Phước
2 Tây Ninh
3 Bình Dương
4 Đồng Nai
5 Bà Rịa - Vũng Tàu
6 TP. Hồ Chí Minh
Mekong river delta
1 Long An
2 Tiền Giang
3 Bến Tre
4 Trà Vinh
5 Vĩnh Long
6 Đồng Tháp
7 An Giang
8 Kiên Giang
9 Cần Thơ

724

30423

42.020718

651

28795

54
126
1853

183
111
450
565

847
1908
54774
2206
1163
14021
8976

15.685185
15.142857

53
130
1178
90
100
457

558
3500
45317
1539
1415
13278

544

28408

52.220588

531

29085

15381
97
84
761
759

549900
766
805
20824
19381

7.8969072
9.5833333
27.363995
25.534914

13720
109
77

527
607

447998
952
662
13409
19558

251

5171

20.601594

335

7808

13429

502953

37.452752

12065

405609

4239

84
215
178
216
572

103312
1295
3622
1506
5072
12563

15.416667
16.846512
8.4606742
23.481481
21.963287

5101
77
315
170
413
853

113450
1309
4940
1559

5179
12834

438

15400

35.159817

344

10785

384

8327

21.684896

482

8360

331

2766

8.3564955

384

3226

1523

47008

30.865397

1662

57411

12.054645
10.477477
31.157778
15.886726

ix

44.23195
1
10.528302
26.923077
17.1
14.15
29.054705
54.77401
1
8.733945

8.5974026
25.444023
32.220758
23.30746
3
33.61864
9
17
15.68254
9.1705882
12.539952
15.045721
31.35174
4
17.34439
8
8.401041
7
34.54332

370

6287

16.991892

852
53
125
1271

190
103
491

30733
446
3243
49400
2984
1570
15761

36.071596
8.4150943

487

29085

59.722793

15318
105
77
883
684

485285
879
904

15529
25987

8.3714286
11.74026
17.586636
37.99269

304

7684

25.276316

13265

434302

32.740445

5273
161
325
166
472
469

123067
3762
5879

1803
5535
14212

23.36646
18.089231
10.861446
11.726695
30.302772

412

12321

29.90534

514

10767

20.947471

380

4221

11.107895

1816

53766

29.606828

38.867034
15.705263
15.242718
32.099796


Case study - ANOVA
10 Hậu Giang
11 Sóc Trăng
12 Bạc Liêu
13 Cà Mau

43

797

18.534884

48

1326

105

2097

19.971429

156

2784

101

2083

20.623762

101

2557

49

776

15.836735

96

1180

x

1
27.625

17.84615
4
25.31683
2
12.29166
7

126

3625

28.769841

171

2989

17.479532

170

2546

14.976471

91

1641

18.032967