Mathematics AI SL
Sample Internal Assessment

Table of content
Rationale
Aim
Research question
Introduction
Data collection
Processed data

Calculation of R2

Calculation of pearson’s correlation coefficient
Conclusion
Bibliography

To what extent is there a Co-relation 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?

1,937 Words
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Word count: 1,937

Rationale

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.

Aim

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.

Research question

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?

Introduction

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.

Data collection

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
Figure 1 - Table On Number Of Individuals Getting Infected By COVID-19 Out Of The Individuals Those Were Going Out For Work Of The Age Group 25 Years To 35 Years
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
Figure 2 - Table On Number Of Individuals Getting Infected By COVID-19 Out Of The Individuals Those Were Going Out For Work Of The Age Group 35 Years To 45 Years
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
Figure 3 - Table On Number Of Individuals Getting Infected By COVID-19 Out Of The Individuals Those Were Going Out For Work Of The Age Group 45 Years To 55 Years

Processed data

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
Figure 4 - Table On Cumulative Frequency Table For The Age Group Of 25 Years To 35 Years
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
Figure 5 - Table On Cumulative Frequency Table For The Age Group Of 35 Years To 45 Years
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
Figure 6 - Table On Cumulative Frequency Table For The Age Group Of 45 Years To 55 Years

Graphical analysis

Figure 7 - Cumulative Number Of Individuals Getting Infected By COVID-19 With Respect To Cumulative Individuals Those Were Going Out For Work Of The Age Group 25 Years To 35 Years
Figure 8 - Number Of Individuals Getting Infected By COVID-19 With Respect To The Individuals Those Were Going Out For Work Of The Age Group 25 Years To 35 Years
Figure 9 - Cumulative Number Of Individuals Getting Infected By COVID-19 With Respect To Cumulative Individuals Those Were Going Out For Work Of The Age Group 35 Years To 45 Years
Figure 10 - Number Of Individuals Getting Infected By COVID-19 With Respect To The Individuals Those Were Going Out For Work Of The Age Group 35 Years To 45 Years
Figure 11 - Cumulative Number Of Individuals Getting Infected By COVID-19 With Respect To Cumulative Individuals Those Were Going Out For Work Of The Age Group 45 Years To 55 Years
Figure 12 - Number Of Individuals Getting Infected By COVID-19 With Respect To The Individuals Those Were Going Out For Work Of The Age Group 45 Years To 55 Years

Calculation of R2

x

y

x2

y2

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

∑x2 = 66084300

∑y2 = 1088363

∑xy = 8468370

Figure 13 - Table On Processed Data For Calculation Of Correlation Coefficient R2 For Group 1 (25 Years To 35 Years)

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

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

=> r2 = 0.9894

x

y

x2

y2

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= 66084300

∑y2 = 118875

∑xy = 2794140

Figure 14 - Table On Processed Data For Calculation Of Correlation Coefficient R2 For Group 2 (35 Years To 45 Years)

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

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

=> r2 = 0.977

x

y

x2

y2

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= 66084300

∑y2 = 1818533

∑xy = 10953790

Figure 15 - Table On Processed Data For Calculation Of Correlation Coefficient R2 For Group 2 (45 Years To 55 Years)

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

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

=> r2 = 0.9968

Calculation of pearson’s correlation coefficient

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

Figure 16 - Table On Processed Data Table For Calculation Of Pearson’s Correlation Coefficient In Group 1

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

=> 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
Figure 17 - Table On Processed Data Table For Calculation Of Pearson’s Correlation Coefficient In Group 2

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 .

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

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

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
Figure 18 - Table On Processed Data Table for calculation of Pearson’s Correlation Coefficient in Group 3

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 .

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

$$=> R=\frac{2363023.5}{\sqrt{15108055370774.95}}$$

R = 0.998

Conclusion

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 = 0.1219x + 12.976

The value of R2 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 = 0.0439x - 3.2908

The value of R2 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 = 0.1564x + 19.344

The value of R2 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.

Bibliography

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• India Coronavirus: 8,041,051 Cases and 120,583 Deaths - Worldometer.https://www.worldometers.info/coronavirus/country/india/. Accessed 29 Oct. 2020.
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