Mathematics AI SL

Sample Internal Assessment

Table of content

Rationale

Aim

Research question

Introduction

Data collection

Processed data

Calculation of R^{2}

Calculation of pearson’s correlation coefficient

Conclusion

Bibliography

10 mins Read

1,937 Words

The world has turned upside down after the spread of corona virus. We all are fighting with the virus in our own ways. To practice social distancing, the world had gone to a lockdown.

It has been months. No school, no outings, no vacations, no playing in parks; we are confined to our houses. The world could not be locked down for months, so with all safety measures it is trying to get back to its workings.

With every passing day, there is so much deaths and active corona cases. It is really disheartening. Though there is no more lockdown, we prefer staying at home. We do not go out until any need.

With everything getting back to normal, a news of reopening schools was surfacing for quite some time for which parents' consent was required.

The decision of going to school required a lot of research because that could not only cost me and my friends and family getting infected but also could cost our lives.

I went on to research more about the spread of the virus. I read many articles, surfed the internet and got to know many factors, precautions and many more. I could not get the answer I was looking for. However, reading several newspapers and statistical data that has been reported collecting data from a number of companies in the cities has helped to get a clear idea about the number of individuals who were bound to go out of their houses to their work places after remission of lockdown period.

I wanted to know to what extent there is a relation between the number of individuals going out to their workplaces after lockdown and the number of individuals getting infected by COVID – 19.

This IA is about the same. This research would allow my family to decide if I should attend school or should I continue with my online classes.

The main motive of this IA is to show a correlation between the number of employees and working professionals those who are getting infected by COVID-19 with respect to the number of working professionals going out right after calling off the Lockdown. The study will show the effect of Lockdown during the COVID-19 pandemic on the rate of spreading after remission of it. It will also redirect to the fact how much it is safe for individuals to go out for work after Lockdown.

What is the relationship between the number of individuals who are going out to their workplaces after Lockdown and the number of individuals who are getting infected by COVID-19?

COVID – 19 was one of the most infectious diseases the world has ever seen. It has spread throughout the globe within a span of 3 to 5 months. Originating from the city of Yuhan in China, it has spread to different countries including the far west – The USA. One of the most affect countries is India. From the month of March to July, complete Lockdown was called by the government to cope up with the pandemic and restrict the spread. However, lockdown was called off by the government since August and emergency services, such as, banking sector, media, etc. were allowed to start their usual operation. In this process, the infection of COVID – 19 is again taking its pace in an increasing order.

In this IA, data of the number of attendees of 20 companies situated in Mumbai, Delhi, and Kolkata, the three most affected cities of India, has been collected and the number of employees who got infected were noted. This collection is done for three different age groups from 25 years old to 55 years old with an interval of 10 years. This will help to analyze the spread of infection among individuals of different age groups after lockdown.

The data collection for this correlative analysis has been done from a number of internationalized media houses and newspaper which are considered as authentic sources of information.

Sl. No.

Total number of individuals

Number of individuals infected

1

150

21

2

160

18

3

150

30

4

120

12

5

200

21

6

140

18

7

160

20

8

150

40

9

170

15

10

150

21

11

120

6

12

160

8

13

150

30

14

160

31

15

120

13

16

150

10

17

140

7

18

180

17

19

160

6

20

120

20

Sl. No.

Total number of individuals

Number of individuals infected

1

150

4

2

160

6

3

150

9

4

120

3

5

200

8

6

140

10

7

160

0

8

150

0

9

170

12

10

150

3

11

120

30

12

160

1

13

150

5

14

160

3

15

120

1

16

150

9

17

140

4

18

180

7

19

160

6

20

120

3

Sl. No.

Total number of individuals

Number of individuals infected

1

150

34

2

160

36

3

150

32

4

120

24

5

200

21

6

140

3

7

160

25

8

150

28

9

170

32

10

150

34

11

120

21

12

160

14

13

150

25

14

160

23

15

120

20

16

150

21

17

140

22

18

180

27

19

160

34

20

120

21

Sl. No.

Cumulative total number of individuals

Cumulative number of infected individuals

1

150

21

2

310

39

3

460

69

4

580

81

5

780

102

6

920

120

7

1080

140

8

1230

180

9

1400

195

10

1550

216

11

1670

222

12

1830

230

13

1980

260

14

2140

291

15

2260

304

16

2410

314

17

2550

321

18

2730

338

19

2890

344

20

3010

364

Sl. No.

Cumulative total number of individuals

Cumulative number of infected individuals

1

150

4

2

310

10

3

460

19

4

580

22

5

780

30

6

920

40

7

1080

40

8

1230

40

9

1400

52

10

1550

55

11

1670

85

12

1830

86

13

1980

91

14

2140

94

15

2260

95

16

2410

104

17

2550

108

18

2730

115

19

2890

121

20

3010

124

Sl. No.

Cumulative total number of individuals

Cumulative number of infected individuals

1

150

34

2

310

70

3

460

102

4

580

126

5

780

147

6

920

150

7

1080

175

8

1230

203

9

1400

235

10

1550

269

11

1670

290

12

1830

304

13

1980

329

14

2140

352

15

2260

372

16

2410

393

17

2550

415

18

2730

442

19

2890

476

20

3010

497

Calculation of R^{2}

*x*

*y*

*x*^{2}

*y*^{2}

*xy*

150

21

22500

441

3150

310

39

96100

1521

12090

460

69

211600

4761

31740

580

81

336400

6561

46980

780

102

608400

10404

79560

920

120

846400

14400

110400

1080

140

1166400

19600

151200

1230

180

1512900

32400

221400

1400

195

1960000

38025

273000

1550

216

2402500

46656

334800

1670

222

2788900

49284

370740

1830

230

3348900

52900

420900

1980

260

3920400

67600

514800

2140

291

4579600

84681

622740

2260

304

5107600

92416

687040

2410

314

5808100

98596

756740

2550

321

6502500

103041

818550

2730

338

7452900

114244

922740

2890

344

8352100

118336

994160

3010

364

9060100

132496

1095640

∑x = 31930

∑y = 4151

∑x^{2} = 66084300

∑y^{2} = 1088363

∑xy = 8468370

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

*r *= \(\frac{20×8468370-(31930)(4151)}{[20×66084300-(31930)^2][20×1088363-(4151)^2]}\)

*=> r*^{2}* *= 0.9894

*x*

*y*

*x*^{2}

*y*^{2}

*xy*

150

4

22500

16

600

310

10

96100

100

3100

460

19

211600

361

8740

580

22

336400

484

12760

780

30

608400

900

23400

920

40

846400

1600

36800

1080

40

1166400

1600

43200

1230

40

1512900

1600

49200

1400

52

1960000

2704

72800

1550

55

2402500

3025

85250

1670

85

2788900

7225

141950

1830

86

3348900

7396

157380

1980

91

3920400

8281

180180

2140

94

4579600

8836

201160

2260

95

5107600

9025

214700

2410

104

5808100

10816

250640

2550

108

6502500

11664

275400

2730

115

7452900

13225

313950

2890

121

8352100

14641

349690

3010

124

9060100

15376

373240

∑x = 31930

∑y = 1335

∑x^{2 }= 66084300

∑y^{2} = 118875

∑xy = 2794140

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

=> *r *= \(\frac{20×2794140-(31930)(1335)}{\sqrt{[20×66084300-(31930)^2][20×118875-(1335)^2]}}\)

*=> r*^{2}* *= 0.977

*x*

*y*

*x*^{2}

*y*^{2}

*xy*

150

34

22500

1156

5100

310

70

96100

4900

21700

460

102

211600

10404

46920

580

126

336400

15876

73080

780

147

608400

21609

114660

920

150

846400

22500

138000

1080

175

1166400

30625

189000

1230

203

1512900

41209

249690

1400

235

1960000

55225

329000

1550

269

2402500

72361

416950

1670

290

2788900

84100

484300

1830

304

3348900

92416

556320

1980

329

3920400

108241

651420

2140

352

4579600

123904

753280

2260

372

5107600

138384

840720

2410

393

5808100

154449

947130

2550

415

6502500

172225

1058250

2730

442

7452900

195364

1206660

2890

476

8352100

226576

1375640

3010

497

9060100

247009

1495970

∑x = 31930

∑y = 5381

∑x^{2 }= 66084300

∑y^{2} = 1818533

∑xy = 10953790

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

=> *r *= \(\frac{20×10953790-(31930)(5381)}{\sqrt{[20×66084300-(31930)^2][20×1818533-(5381)^2]}}\)

*=> r*^{2}* *= 0.9968

*x*

*y*

\( x-\bar x\)

\(y-\bar y\)

\((x-\bar x)(y-\bar y)\)

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

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

150

21

1596.5

207.55

-1446.5

-186.55

269844.575

310

39

1596.5

207.55

-1286.5

-168.55

216839.575

460

69

1596.5

207.55

-1136.5

-138.55

157462.075

580

81

1596.5

207.55

-1016.5

-126.55

128638.075

780

102

1596.5

207.55

-816.5

-105.55

86181.575

920

120

1596.5

207.55

-676.5

-87.55

59227.575

1080

140

1596.5

207.55

-516.5

-67.55

34889.575

1230

180

1596.5

207.55

-366.5

-27.55

10097.075

1400

195

1596.5

207.55

-196.5

-12.55

2466.075

1550

216

1596.5

207.55

-46.5

8.45

-392.925

1670

222

1596.5

207.55

73.5

14.45

1062.075

1830

230

1596.5

207.55

233.5

22.45

5242.075

1980

260

1596.5

207.55

383.5

52.45

20114.575

2140

291

1596.5

207.55

543.5

83.45

45355.075

2260

304

1596.5

207.55

663.5

96.45

63994.575

2410

314

1596.5

207.55

813.5

106.45

86597.075

2550

321

1596.5

207.55

953.5

113.45

108174.575

2730

338

1596.5

207.55

1133.5

130.45

147865.075

2890

344

1596.5

207.55

1293.5

136.45

176498.075

3010

364

1596.5

207.55

1413.5

156.45

221142.075

*Calculation:*

\(\bar x=\frac{∑x}{N}=\frac{31930}{20}=1596.5\)

\(\bar y=\frac{∑y}{N}=\frac{4151}{20}=207.55\)

\(∑(x-\bar x)(y-\bar y)=1841298.5\)

\(∑(x-\bar x)^2=15108055\)

\(∑(y-\bar y)^2=226822.95\)

Let, the Pearson’s Correlation Coefficient be ℜ.

R = \(\frac{∑(x-\bar x)(y-\bar y)}{∑(x-\bar x)^2\times∑(y-\bar y)^2}\)

=> R = \(\frac{1841298.5}{15108055226822.95}\)

R = 0.998

*x*

*y*

\( x-\bar x\)

\(y-\bar y\)

\((x-\bar x)(y-\bar y)\)

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

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

150

4

1596.5

66.75

-1446.5

-62.75

90767.875

310

10

1596.5

66.75

-1286.5

-56.75

73008.875

460

19

1596.5

66.75

-1136.5

-47.75

54267.875

580

22

1596.5

66.75

-1016.5

-44.75

45488.375

780

30

1596.5

66.75

-816.5

-36.75

30006.375

920

40

1596.5

66.75

-676.5

-26.75

18096.375

1080

40

1596.5

66.75

-516.5

-26.75

18096.375

1230

40

1596.5

66.75

-366.5

-26.75

9803.875

1400

52

1596.5

66.75

-196.5

-14.75

2898.375

1550

55

1596.5

66.75

-46.5

-11.75

546.375

1670

85

1596.5

66.75

73.5

18.25

1341.375

1830

86

1596.5

66.75

233.5

19.25

4494.875

1980

91

1596.5

66.75

383.5

24.25

9299.875

2140

94

1596.5

66.75

543.5

27.25

14810.375

2260

95

1596.5

66.75

663.5

28.25

18743.875

2410

104

1596.5

66.75

813.5

37.25

30302.875

2550

108

1596.5

66.75

953.5

41.25

39331.875

2730

115

1596.5

66.75

1133.5

48.25

54691.375

2890

121

1596.5

66.75

1293.5

54.25

70172.375

3010

124

1596.5

66.75

1413.5

57.25

80922.875

Calculation

\(\bar x=\frac{\sum x}{N}=\frac{31930}{20}\) = 1596.5

\(\bar y=\frac{\sum y}{N}=\frac{1335}{20}\) = 66.75

\(\sum(x-\bar x)(y-\bar y)=662812.5\)

\(\sum(x-\bar x)^2=15108055\)

\(\sum(y-\bar y)^2=29763.75\)

Let, the Pearson’s Correlation Coefficient be ℜ.

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

\(=>R=\frac{662812.5}{\sqrt{15108055\times29763.75}}\)

*R* = 0.988

*x*

*y*

\(x-\bar x\)

\(y-\bar y\)

\((x-\bar x)(y-\bar y)\)

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

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

150

34

1596.5

269.05

-1446.5

-235.05

339999.825

310

70

1596.5

269.05

-1286.5

-199.05

256077.825

460

102

1596.5

269.05

-1136.5

-167.05

189852.325

580

126

1596.5

269.05

-1016.5

-143.05

145410.325

780

147

1596.5

269.05

-816.5

-122.05

99653.825

920

150

1596.5

269.05

-676.5

-119.05

80537.325

1080

175

1596.5

269.05

-516.5

-94.05

48576.825

1230

203

1596.5

269.05

-366.5

-66.05

24207.325

1400

235

1596.5

269.05

-196.5

-34.05

6690.825

1550

269

1596.5

269.05

-46.5

-0.05

2.325

1670

290

1596.5

269.05

73.5

20.95

1539.825

1830

304

1596.5

269.05

233.5

34.95

8160.825

1980

329

1596.5

269.05

383.5

59.95

22990.825

2140

352

1596.5

269.05

543.5

82.95

45083.325

2260

372

1596.5

269.05

663.5

102.95

68307.325

2410

393

1596.5

269.05

813.5

123.95

100833.325

2550

415

1596.5

269.05

953.5

145.95

139163.325

2730

442

1596.5

269.05

1133.5

172.95

196038.825

2890

476

1596.5

269.05

1293.5

206.95

267689.825

3010

497

1596.5

269.05

1413.5

227.95

322207.325

Calculation

\(\bar x=\frac{\sum x}{N}=\frac{31930}{20}\) = 1596.5

\(\bar y=\frac{\sum y}{N}=\frac{5381}{20}\)= 269.05

\(\sum(x-\bar x)(y-\bar y)=2363023.5\)

\(\sum(x-\bar x)^2=15108055\)

\(\sum(y-\bar y)^2=370774.95\)

Let, the Pearson’s Correlation Coefficient be ℜ.

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

\(=> R=\frac{2363023.5}{\sqrt{15108055370774.95}}\)

*R *= 0.998

In this IA, a correlation has been developed between the total number of individuals who are going to their respective work places after calling off of Lockdown in India and thereby getting infected by COVID-19. For the first group, i.e., between the age of 25 years and 35 years, working individuals are getting infected by COVID-19 in an increasing fashion. The trendline shows an increasing direct relationship between the number of individuals going out for their work and the ones those who are getting infected. The equation of the trendline is shown below:

*y* = 0.1219*x* + 12.976

The value of R^{2} correlation coefficient for the graph is 0.9894 which validates the fact that the relation is linear and increasing. Furthermore, the value of Pearson’s Correlation Coefficient for this graph is 0.998. As the value is very close to 1, it justifies the claim that the relationship is linear and being a positive value, it satisfies the claim that the relationship is increasing or direct. The reason behind such a correlation is assumed to be the work load that is there on the individuals of this age group. On the other hand, being on the lower side of the age group, their immunity is comparatively less strong than that of the other age groups. Thus, the spread of infection is taking a significant number in this age group.

For the second group, i.e., between the age of 35 years and 45 years, working individuals are getting infected by COVID-19 is showing a direct increasing relation. The equation of the trendline is shown below:

*y* = 0.0439*x* - 3.2908

The value of R^{2} correlation coefficient for the graph is 0.977 which validates the fact that the relation is linear and increasing. Furthermore, the value of Pearson’s Correlation Coefficient for this graph is 0.988. As the value is very close to 1, it justifies the claim that the relationship is linear and being a positive value, it satisfies the claim that the relationship is increasing or direct. The spread is comparatively less in this group as the immunity of the individuals of this age group is considerably more and with an increase in age, people often tend to be more cautious and aware and take significant preventive measures to protect them from the disease.

Finally, in the third group, i.e., between the age of 45 years and 55 years, working individuals are getting infected by COVID-19 is showing a direct increasing relation. The equation of the trendline is shown below:

*y *= 0.1564*x* + 19.344

The value of R^{2} correlation coefficient for the graph is 0.9968 which validates the fact that the relation is linear and increasing. Furthermore, the value of Pearson’s Correlation Coefficient for this graph is 0.998. As the value is very close to 1, it justifies the claim that the relationship is linear and being a positive value, it satisfies the claim that the relationship is increasing or direct. This group has shown maximum infection than that of the other groups. It is due to the fact that, this age group is quite close to the limit of senior citizens. According to the doctors, people aged more than 50 are more vulnerable to the disease which is significantly proved in its correlative study.

- Novel, Coronavirus Pneumonia Emergency Response Epidemiology. "The epidemiological characteristics of an outbreak of 2019 novel coronavirus diseases (COVID-19) in China." Zhonghua liu xing bing xue za zhi= Zhonghua liuxingbingxue zazhi 41.2 (2020): 145.
*India Coronavirus: 8,041,051 Cases and 120,583 Deaths - Worldometer.*https://www.worldometers.info/coronavirus/country/india/. Accessed 29 Oct. 2020.- “How Does Coronavirus Spread?” WebMD,https://www.webmd.com/lung/coronavirus-transmission-overview. Accessed 29 Oct. 2020.
- Desk, The Hindu Net. “Coronavirus India Lockdown Day 169 Updates | September 10, 2020.” The Hindu, 10 Sept. 2020. www.thehindu.com,https://www.thehindu.com/news/national/coronavirus-india-lockdown-september-10-2020-live-updates/article32567344.ece.
- “Covid 19 Lockdown: How India Compares to Other Coronavirus Hotbeds.” The Economic Times. The Economic Times,https://economictimes.indiatimes.com/news/politics-and-nation/lockdown-stats-how-india-compares-to-other-coronavirus-hotbeds/articleshow/74805295.cms. Accessed 6 Nov. 2020.
- “India - Online Deliveries after COVID-19 Lockdown 2020.” Statista,https://www.statista.com/statistics/1115652/india-coronavirus-post-lockdown-purchase-e-commerce/. Accessed 6 Nov. 2020.
- Benesty, Jacob, et al. "Pearson correlation coefficient." Noise reduction in speech processing. Springer, Berlin, Heidelberg, 2009. 1-4