Mathematics AI SL

Sample Internal Assessment

Table of content

Rationale

Aim

Introduction

Data collection

Calculation of R^{2} for graph 1

Calculation of pearson’s correlation coefficient for graph 1

Conclusion

Bibliography

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I have grown up aspiring to take up degree courses in abroad in the top colleges and have been preparing accordingly. Higher studies in abroad is a big deal for everyone in my family as no-one has done it before. Despite initial dilemma, everyone quite encouraged and supported me.

Recently I came across a statement while going through multiple reviews. The statement read that family income is a factor which determines the SAT score.

SAT is an entrance examination which is necessary for a bachelor's degree in abroad.

So, I started my research regarding the same and tried to find the correlation between the SAT score of an aspirant and his or her family income. This IA is based on this correlation.

The main motive of this IA is to cite a relationship between SAT score and average family income of score holder. In addition to that, a regression model will be prepared in this IA on same topic.

The SAT is an entrance exam used by most colleges and universities to make admissions decisions. The SAT is a multiple-choice, pencil-and-paper test created and administered by the College Board. The purpose of the SAT is to measure a high school student's readiness for college, and provide colleges with one common data point that can be used to compare all applicants. College admissions officers will review standardized test scores alongside your high school GPA, the classes you took in high school, letters of recommendation from teachers or mentors, extracurricular activities, admissions interviews, and personal essays. How important SAT scores are in the college application varies.

Overall, the higher you score on the SAT and/or ACT, the more options for attending and paying for college will be available to you.

SAT is considered as one of the most expensive exams to take because of several reasons. Firstly, SAT is an internationally acclaimed examination. Thus, the registration cost is in USD. Currently, the registration fee of SAT is $40 or $60. Secondly, the syllabus of SAT differs completely from the syllabus of school academics in India. Thus, separate coaching is very necessary for SAT which in turn is very expensive in almost every country.

A survey has been carried on in which SAT score and average income of score holders are noted. The surveyed data is shown below. All the data is shown in ascending order for better understanding:

Serial No.

Country

SAT Score

Annual Income (in lakh in INR)

1

Bangladesh

900

3.5

2

Sri Lanka

906

3.5

3

India

907

3.5

4

Bangladesh

910

3.5

5

India

916

3.5

6

Sri Lanka

919

4

7

Bangladesh

928

4

8

India

939

4

9

Singapore

940

4

10

India

945

4

11

Sri Lanka

948

4.5

12

Bangladesh

955

4.5

13

Germany

970

4.5

14

India

971

4.5

15

Sri Lanka

971

4.5

16

Austria

972

5

17

Bangladesh

973

5

18

India

973

5

19

Canada

973

5

20

Sri Lanka

973

5

21

Austria

978

5.5

22

Singapore

979

5.5

23

Bangladesh

980

5.5

24

India

985

5.5

25

Bangladesh

986

5.5

26

United States

990

6

27

India

991

6

28

Sri Lanka

993

6

29

United Kingdom

996

6

30

Bangladesh

996

6

31

India

997

6.5

32

United Kingdom

997

6.5

33

India

998

6.5

34

Sri Lanka

999

6.5

35

India

1001

6.5

36

Bangladesh

1002

7

37

Australia

1005

7

38

Sri Lanka

1006

7

39

Singapore

1006

7

40

India

1006

7

41

Australia

1006

7.5

42

India

1009

7.5

43

Bangladesh

1015

7.5

44

Germany

1016

7.5

45

Singapore

1019

7.5

46

India

1019

8

47

Austria

1019

8

48

Bangladesh

1019

8

49

Germany

1020

8

50

Sri Lanka

1021

8

51

India

1025

8.5

52

India

1025

8.5

53

Sri Lanka

1025

8.5

54

Germany

1025

8.5

55

India

1025

8.5

56

United States

1025

9

57

India

1025

9

58

Hong Kong

1025

9

59

Germany

1039

9

60

Australia

1040

9

61

India

1042

9.5

62

Germany

1042

9.5

63

Australia

1043

9.5

64

Germany

1043

9.5

65

China

1043

9.5

66

India

1046

10

67

Australia

1050

10

68

Germany

1050

10

69

Austria

1050

10

70

Germany

1050

10

71

United States

1050

10.5

72

India

1050

10.5

73

United States

1050

10.5

74

Germany

1056

10.5

75

United States

1056

10.5

76

Singapore

1060

11

77

Singapore

1069

11

78

United States

1069

11

79

Germany

1070

11

80

India

1075

11

81

United States

1076

11.5

82

France

1079

11.5

83

Germany

1080

11.5

84

India

1081

11.5

85

United Kingdom

1085

11.5

86

Germany

1086

12

87

India

1087

12

88

United States

1088

12

89

Canada

1088

12

90

Singapore

1088

12

91

Germany

1100

12.5

92

Germany

1105

12.5

93

India

1109

12.5

94

United States

1120

12.5

95

France

1125

12.5

96

France

1126

13

97

Canada

1126

13

98

United Kingdom

1129

13

99

Singapore

1130

13

100

United States

1135

13

Sample Calculation:

Mean = \(\frac{y_1+y_2+y_3+y_4+y_5}{5}\)

Mean Score of Group 1 = \(\frac{900+906+907+910+916}{5}\)=907.8

Standard Deviation = \(\sqrt{\frac{(\bar y-y_1)^2+(\bar y-y_2)^2+(\bar y-y_3)^2+(\bar y-y_4)^2+(\bar y-y_5)^2}{5}}\)

SD of Group 1 = \(\sqrt{\frac{(907.8-900)^2+(907.8-906)^2+(907.8-907)^2+(907.8-910)^2+(907.8-916)^2}{5}}\)=5.23

Mode = 1025 and 1050

c

y

x^{2}

y^{2}

xy

3.5

907.8

12.25

824100.8

3177.3

4

934.2

16

872729.6

3736.8

4.5

963

20.25

927369

4333.5

5

972.8

25

946339.8

4864

5.5

981.6

30.25

963538.6

5398.8

6

993.2

36

986446.2

5959.2

6.5

998.4

42.25

996802.6

6489.6

7

1005

49

1010025

7035

7.5

1013

56.25

1026169

7597.5

8

1019.6

64

1039584

8156.8

8.5

1025

72.25

1050625

8712.5

9

1030.8

81

1062549

9277.2

9.5

1042.6

90.25

1087015

9904.7

10

1049.2

100

1100821

10492

10.5

1052.4

110.25

1107546

11050.2

11

1068.6

121

1141906

11754.6

11.5

1080.2

132.25

1166832

12422.3

12

1087.4

144

1182439

13048.8

12.5

1111.8

156.25

1236099

13897.5

13

1129.2

169

1275093

14679.6

∑x=165

∑y=20465.8

Σx^{2}=1527.5

Σy^{2}=21004028

Σxy=171987.9

r =\(\frac{n\big(∑ xy\big)-(∑ x)(∑ y)}{\sqrt{[n∑ x^2-\big(∑ x\big)^2][n∑ y^2-\big(∑ y\big)^2]}}\)

=> r =\(\frac{20(171987.9)-(165)(20465.8)}{\sqrt{[20×1527.5-(165)^2][20×21004028-(20465.8)^2]}}\)

=> r = 0.9829

=> r^{2 }= 0.9662

x

y

\(x-\bar x\)

\(y\,-\bar{y}\)

\((x-\bar{x})(y-\bar{y})\)

\((x-\bar{x})^2\)

\((y-\bar{y})^2\)

3.5

907.8

-4.75

-115.49

548.5775

22.5625

13337.94

4

934.2

-4.25

-89.09

378.6325

18.0625

7937.028

4.5

963

-3.75

-60.29

226.0875

14.0625

3634.884

5

972.8

-3.25

-50.49

164.0925

10.5625

2549.24

5.5

981.6

-2.75

-41.69

114.6475

7.5625

1738.056

6

993.2

-2.25

-30.09

67.7025

5.0625

905.4081

6.5

998.4

-1.75

-24.89

43.5575

3.0625

619.5121

7

1005

-1.25

-18.29

22.8625

1.5625

334.5241

7.5

1013

-0.75

-10.29

7.7175

0.5625

105.8841

8

1019.6

-0.25

-3.69

0.9225

0.0625

13.6161

8.5

1025

0.25

1.71

0.4275

0.0625

2.9241

9

1030.8

0.75

7.51

5.6325

0.5625

56.4001

9.5

1042.6

1.25

19.31

24.1375

1.5625

372.8761

10

1049.2

1.75

25.91

45.3425

3.0625

671.3281

10.5

1052.4

2.25

29.11

65.4975

5.0625

847.3921

11

1068.6

2.75

45.31

124.6025

7.5625

2052.996

11.5

1080.2

3.25

56.91

184.9575

10.5625

3238.748

12

1087.4

3.75

64.11

240.4125

14.0625

4110.092

12.5

1111.8

4.25

88.51

376.1675

18.0625

7834.02

13

1129.2

4.75

105.91

503.0725

22.5625

11216.93

Calculation

\(\bar{x}=\frac{\sum x}{20}=\frac{165}{20}\) 8.25

\(\bar y=\frac{\sum y}{20}=\frac{20465.8}{20}\) =1023.29

\(\sum(x-\bar x)(y-\bar y)\) =3145.05

\(\sum(x-\bar x)^2\) =166.25

\(\sum(y-\bar y)^2\) =61579.8

Let, the Pearson’s Correlation Coefficient be ℜ.

R= \(\frac{\sum(x-\bar x)(y-\bar y)}{\sqrt{\sum(x-\bar x)^2×\sum (y-\bar y)^2}}\)

R= \(\frac{3145.05}{\sqrt{166.25×61579.8}}=\frac{3145.05}{\sqrt{10237641.75}}=\frac{3145.05}{3199.63}\)

R=0.982

In this IA, I have deduced a relationship between SAT Scores and annual income of score holder. From the background information study, we have found that, SAT is one of the few entrance examinations that requires the SAT aspirant to be financially stable. From the collected data, we have concluded that, with increase in annual family income of candidates, the aberration in marks achieved amongst the candidates of each income group tends to decrease though there are some exceptions. The exceptions in getting a high range of marks secured by the candidates of higher income group may get nullified by taking considerably large data sheet. Though, in some cases, with increased family income. Nowadays, tendency of securing in-depth knowledge on any topic seems to decrease amongst the students belonging to such groups. But with increase in family income, usually, the range of marks achieved is decreasing and often in some groups, the score of all the candidates is same because of getting almost same intensity of tutorial or guidance from several institutes as well as study materials. Furthermore, in low income groups, standard deviation is more because of lack of availability of traditional guidance required for SAT examination. The median of each of the income groups lie close to the mean value of SAT score which signifies that the marks secured by the candidates of each group are very close to each other. On the other hand, in the survey of 100 candidates, 7 candidates have secured 1025 and 1050 score. Thus, it can be stated that the frequency of these two scores is maximum and most of the candidates are likely to secure a score which is equal to 1025 and 1050 or close to it. Thus, the mode of the data sheet is 1025 and 1050. From the above survey, we have concluded the graph that shows a positive increasing relationship. Initially, I have derived a linear relationship using the collected data. The equation of the relationship is given by:

y = 0.0507x + 43.6

R^{2} = 0.9661

From this data, we can clearly say that, with increase in family income, the candidates are being able to get more efficient and professional tutorials as well as study materials which helps the candidates in boosting their SAT score. The correlation co-efficient is also 0.9661 which is very close to 1, which validates our conclusion.

In addition, we have found the Pearson’s Correlation coefficient to establish another correlation analysis giving more validation to this IA. In Pearson’s Correlation, we know that the coefficient lies between 1 and -1 where 1 positive side signifies direct relationship between the two variables and negative side signifies inverse or indirect relationship between the two variables. In this correlation, zero signifies no relationship. In this IA, the value of Pearson’s correlation constant has come out to be 0.982 which is very close to 1 signifying a positive relationship between SAT score and the average family income of the candidates with a strength of very close to 1. Thus, it proves that, the correlation is also linear in nature.

- Bagamery, Bruce D., John J. Lasik, and Don R. Nixon. "Determinants of success on the ETS Business Major Field Exam for students in an undergraduate multisite regional university business program." Journal of Education for Business 81.1 (2005): 55-63.
- https://collegereadiness.collegeboard.org/sat/register/fees
- https://www.theatlantic.com/politics/archive/2014/03/the-real-problem-with-the-sat/453804/
- Benesty, Jacob, et al. "Pearson correlation coefficient." Noise reduction in speech processing. Springer, Berlin, Heidelberg, 2009. 1-4.