Maths IA Final (2019 ) PDF

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IB Maths IA...

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IB Mathematical Studies - Internal Assessment

What is the relationship between the GDP per capita and the value added in the agricultural sector (% of GDP) in 2013?

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Contents Introduction ………………………………………………………………………. …………Page 3 Variables ………………………………………………………………......................... … Page 4 Sampling method ………………………………………………………………………... ...Page 4-5 Sampled Data ……………………………………………………………………………… Page 5-8 Calculations ………………………………………………………………………………… Page 9-12 Scatter Plot Graph ………………………………………………………………. Page 9 Manual Calculation of Pearson’s Correlation Coefficient ……………………. Page 10-12 Least Squares regression ……………………………………………………… Page 12 Conclusion ………………………………………………………………………………… Page 13 Validity ……………………………………………………………………………………… Page 13 Bibliography ………………………………………………………………………………… Page 14 Appendix ………………………………………………………………………………….. Page 15-24

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Introduction Many countries all over the world that are still in the developmental stages feature farming and agriculture as a main way of making an income. Thus impacting the country’s GDP. Having grown up in Thailand for instance I witnessed this first hand, I was also able to observe Thailand develop and slowly introduce new ways of making income from secondary and tertiary industries. I’m interested to see to what degree does agriculture help a countries GDP and whether it is a sustainable way of making an income. Hence for my IA I have decided to examine the value added in the agricultural sector (% of GDP) and the GDP per capita in order to determine if there is a relationship between them. Statement of task The aim of this investigation is to determine whether there is a correlation between value added in the agricultural sector and a countries GDP in 2013. Plan In this IA, the first thing I will do is obtain data from a reliable source. Namely, “Our World in Data” (Our World in Data, 2019). I will then insert the data into an excel worksheet and make a scatter plot graph, I will remove data that does not fit into the category (data that is reflective of a continent/ or level of development – “HICs) I will do this so I can identify the strength of the relationship through the correlation coefficient value accurately. Following this, I will sample the data using a specified sampling method as described below. I will do this so I can obtain a set of data that features absolutely no bias. The next step will be to calculate the correlation manually using excel formulas and a GDC calculator. I will do this so that I can consolidate my knowledge on the mathematical processes, alongside confirming the correlation value originally obtained is accurate. The final process I will be carrying out, is finding the equation of least squares

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regression. I did this by using an excel worksheet, and analyzing the individual components to identify key values.

Variables Independent Variable (𝒚): Value added in the agriculture sector (% of GDP) in 2013. Dependent variable (𝒙): GDP per capita in 2013. Sampling method To begin, I inserted data collected from “Our World in Data” into an excel worksheet. I then went through all the data, removing any elements that were not countries, for example Europe, MEDCs etc.

Figure 1: Example of data that has been removed

After the removal of these values, each piece of data as assigned either the value 1 or 2 systematically. I did this by going through each piece of data assigning the according number (1,2,1,2) as seen below:

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Figure 2: Screenshot from excel worksheet used, displaying how the data was sampled.

Once each piece of data was sampled accordingly, a coin was flipped. If the coin landed on heads, sample one would be used. If the coin landed on tails sample 2 would be used. The outcome of this tails. Thus meaning sample 2, was the set of data that was chosen.

Figure 3: Sampled Data Value added in the GDP per capita (constant

agricultural sector (% of

2011 international $)

GDP) (% of GDP)

Antigua and Barbuda

18823.75445

1.998156663

Argentina

19482.1903

6.052918437

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Australia

43118.08613

2.282509346

Azerbaijan

16593.18876

5.36626448

Bahrain

42680.20134

0.274788371

Barbados

16409.91709

1.441734417

Belgium

40928.13661

0.687975067

Benin

1934.649107

21.34568201

Bhutan

7070.792468

16.10386948

Bosnia and Herzegovina

10284.78455

6.841390006

Brazil

15432.89363

4.506886182

Bulgaria

15997.3568

4.595260089

Cameroon

3100.17131

13.89889921

Cape Verde

5963.907977

8.279068042

China

11951.24796

9.295189652

Costa Rica

14035.28798

5.043542201

Croatia

20271.28013

3.705289401

Czech Republic

28379.74612

2.403634049

Dominican Republic

11907.20486

5.304865946

Egypt

9814.21213

11.2743496

Equatorial Guinea

32736.09743

1.19199068

Fiji

7839.874778

10.11352173

Gabon

16503.06836

3.328922387

Georgia

8254.010766

8.175897646

Ghana

3807.342653

21.65795744

Grenada

11262.6988

4.835402519

Guinea

1711.708994

17.54527212

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Guyana

6696.783715

16.53607542

Honduras

4177.580271

12.19112794

Hungary

23119.01697

3.8520218

Indonesia

9643.27483

13.35669916

Iraq

15223.58271

4.768

Israel

31250.80201

1.196776626

Jamaica

8025.471361

5.974953527

Jordan

8756.710839

2.992380101

Kenya

2683.045833

26.44332979

Kosovo

8584.475807

11.98828763

Kyrgyzstan

3120.542044

14.63787255

Latvia

21563.72143

3.280302431

Lebanon

14400.6408

3.980795251

Macedonia

11877.77368

10.02747609

Malawi

1062.164941

28.67364219

Maldives

13148.73197

5.413434052

Malta

30050.13114

1.190574902

Mauritania

3565.521999

17.99607499

Mexico

16385.05961

3.13878371

Moldova

4541.868222

12.32016715

Montenegro

14551.44671

8.034660111

Mozambique

1034.589423

24.08148941

Namibia

9255.656532

6.344460829

Pakistan

4464.090289

23.83211646

Palestine

4498.302206

4.146361013

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Papua New Guinea

3282.463751

19.25986463

Peru

11430.25578

6.666911913

Poland

23555.49621

2.873929851

Puerto Rico

34913.62274

0.826451928

Romania

19018.87078

5.396890138

Rwanda

1552.751769

28.891371

Saint Lucia

12415.73105

2.356366299

Sao Tome and Principe

2785.734046

12.0339597

Senegal

2195.963854

13.69653049

Seychelles

23540.38038

2.667338884

Slovenia

27622.68555

1.798180665

South Africa

12339.7324

2.097756311

Spain

30678.9173

2.510400285

Suriname

15419.30738

8.68617867

Sweden

43475.80134

1.226289547

Tajikistan

2440.586951

20.43032509

Thailand

14771.46675

11.32486762

Trinidad and Tobago

31405.44627

0.371828624

Turkey

21650.75677

6.725324576

Tuvalu

3143.110114

19.57636686

Ukraine

8338.91505

8.78638928

United Kingdom

37398.79722

0.65373164

Uzbekistan

5067.360265

17.42556563

Venezuela

17665.24

4.902550073

Zambia

3576.680691

8.226563713

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Scatter Plot Creating a scatter plot through excel will enable me to establish the trend between the two variables. I will also be able to identify the strength of the relationship.

Figure 4: Scatter Plot displaying the relationship between GDP per capita and the value added from the agricultural sector.

As can be seen in Figure 1, a strong negative relationship is present between the GDP per capita and the value added from the agricultural sector. Shown in the correlation coefficient, also referred to as the r value, where r = 0.7426843206. Which suggests that the less reliant a country is on the agricultural sector, the higher the GDP per capita. However, I will now identify the strength of the relationship more accurately by calculating the correlation coefficient (r) manually.

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The Pearson Product – moment correlation coefficient is a measure of correlation between two variables X and Y, giving a value between +1 and -1. Its widely used in the sciences as a measure of strength of linear dependence between two variables. (La Rondie, 2012). As we can see from the table below, my correlation coefficient (inster r value) is a moderately strong correlation. Interpreting the R-Value: R value

Correlation

0 < |r| ≤ 0.25

Very weak

0.25 < |r| ≤ 0.5

Weak

0.5 < |r| ≤ 0.75

Moderate

0.75 < |r| ≤ 1

Strong

Manual Calculation of the Correlation Coefficient 𝒓 =%

𝚺( 𝒙 − 𝒙% 𝒚 − 𝒚 ) 𝚺(𝒙 − 𝒙)𝟐 𝚺(𝒚 − 𝒚)𝟐 (Khot, 2019)

A limitation in the maths studies course is the lack of in-depth understanding of what these formulas actually do. Using the TI-84 I was able to calculate a correlation coefficient. Breaking the formula down, I determined that there are 2 very specific sections which are essential for the correlation coefficient calculations. These 2 sections are: 1.

𝒙 − 𝒙%

2. (𝒚 − 𝒚) By exploring these sections, we can see that they both identify the difference between the mean of the 𝑥%or%𝑦 value and the individual 𝑥%or%𝑦 component for each specific country. Finding the

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mean of the 𝑥%or%𝑦 values, named 𝒙%or%𝒚, was relatively simple. Using Excel’s formulas, I was able to find the sum of all my 𝑥%or%𝑦 values and divide by the total number of pieces of data which was in my sample, this was 76 pieces of data. calculated the correlation value manually using an excel spreadsheet where I obtained the following results:

𝒓 =%

𝑥

= 14515.0762

𝑦

= 8.74494849

Σ 𝑥 −𝑥 𝑦−𝑦

= -4717700.316

Σ(𝑥 − 𝑥%)0

= 9675640732

Σ(𝑦 − 𝑦)0

= 4170.384

−4717700.316

= -0.74268125

9675640732 ∗ 4170.384%

Figure 6: Correlation Coefficient calculated manually in Excel (screenshot from excel) As can be seen, the correlation value calculated using the TI-84 and the value obtained through excel is the same. Meaning that the processes used were accurate

Figure 5: Correlation Coefficient calculated on the TI-84

and valid. By calculating the cc manually, I have been

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able to consolidate my knowledge on this individual mathematical process.

Least Squares Regression The origin of the least squares regression method was used to investigate the relationship between the heights of fathers and sons. It was found that the sons tend to regress towards the mean. Using least squares regression, we can create a scatter plot graph in order to identify the data, highlight a mean point and then insert a regression line through the mean point previously identified. In order to improve the line, we can use residuals in the graph. A residual is the vertical distance between a specific data point and the graph of a regression equation. (La Rondie, 2012) 𝑎=

> ?@? A@A

Formula Used

>(?@?%)B

Σ 𝑥 −𝑥 𝑦−𝑦

="-4717700.316"

Σ(𝑥 − 𝑥%)0

="9675640732"

A=

= - 0.000487585 𝑏 = 𝑦 − 𝑎𝑥

B = 8.74494849 – (-0.000487585) (14515.0762) B = 15.82228182

B= 𝑦 = 𝑎𝑥 + 𝑏

Y = -0.00048%𝑥 + 15.8222

Figure 7: Least Squares Regression calculation

As seen above, a negative gradient was obtained through the manual calculation of Least Squares Regression. This calculation tells us the equation of the line as seen on the scatter plot graph in figure 4. To confirm the accuracy of the process, we can substitute an x value in. Take Kosovo as an example:

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Kosovo: 𝑥 = 8584.475807 𝑦 = 11.98828763 𝑦 = 𝑎𝑥 + 𝑏 𝑦 = −0.000487585𝑥 + 15.82228182 𝑦 = −0.000487585%(8584.475807) + 15.82228182 = 11.70165161 The value obtained through the least squares regression process deviates away from the true value by 0.29. which represents a relatively accurate calculation.

Conclusion After creating a scatter plot graph and identifying the correlation coefficient, I was able to highlight the strength of the relationship between value added in the agricultural sector and GDP per capita. It was identified that the r value was equal to 0.7426843206 which represents a moderately strong relationship. I was able to confirm the accuracy of this value by applying a mathematical process, through excel I calculated the correlation manually. Upon which I obtained the same result. As demonstrated in the Least Squares Regression Process I was able to obtain a negative gradient, I also was able to confirm the accuracy of this process by substituting in a X value from my sampled data. As highlighted, the obtained value and the true value were not exact hover they were relatively close. Overall, it can be seen that as value added from the agricultural sector increased overall GDP decreases. Validity Overall, the mathematical processes chosen were reliable and valid tests. The Pearson Correlation-Coefficient allowed me to obtain a value that represents the strength of the relationship between the two variables. However, the correlation value equaled to 0.7426843206. Although this is a moderately strong correlation, an ideal correlation would be >

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0.8 as this value represents a strong correlation. Making interpretations when the correlation is moderate questions the validity of the results. The sample size that I used had 76 individual pieces of data from all over the world, this improved the validity of the investigation because the results can be generalized well. The Least Squares Regression process was not fully accurate as there was some discrepancy between the obtained value and true value however the differences was small enough to maintain accuracy.

Bibliography Our World in Data. (2019). Agriculture value added per worker vs. GDP per capita. [online] Available at: https://ourworldindata.org/grapher/agriculture-value-added-per-worker-vs-gdp-per-capita [Accessed 15 Nov. 2019].

Our World in Data. (2019). GDP per capita. [online] Available at: https://ourworldindata.org/grapher/gdpper-capita-worldbank?year=2013 [Accessed 13 Nov. 2019].

Khot, H. (2019). Correlation Coefficient Formula | Calculation in Excel with Examples. [online] Wallstreet Mojo. Available at: https://www.wallstreetmojo.com/correlation-coefficient-formula/ [Accessed 14 Nov. 2019].

La Rondie, P. (2012). Mathematics standard level. Oxford: Oxford University Press, pp.349-350.

La Rondie, P. (2012). Mathematics standard level. Oxford: Oxford University Press, p.345.

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Statistics Solutions. (2019). Pearson's Correlation Coefficient - Statistics Solutions. [online] Available at: https://www.statisticssolutions.com/pearsons-correlation-coefficient/ [Accessed 6 Oct. 2019].

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Appendix Appendix 1: Raw Data GDP per capita (constant 2011 international $) 1848.700026 10492.80628

Value added in the agricultural sector (% of GDP) 22.58745316 19.56517557

Algeria Antigua and Barbuda Arab World Argentina

13253.61548 18823.75445 15174.1017 19482.1903

9.851117934 1.998156663 5.219866974 6.052918437

Armenia Australia Austria

7727.929092 43118.08613 44301.10403

18.43476288 2.282509346 1.254721216

Azerbaijan Bahamas Bahrain Bangladesh

16593.18876 28871.35826 42680.20134 2835.76662

5.36626448 0.914599721 0.274788371 15.493278

Barbados Belarus Belgium Belize

16409.91709 17656.11934 40928.13661 7856.379109

1.441734417 6.809092448 0.687975067 13.44273318

Benin Bermuda Bhutan Bolivia

1934.649107 50669.31477 7070.792468 6090.69557

21.34568201 0.745266618 16.10386948 9.966979368

Bosnia and Herzegovina Botswana Brazil Brunei Bulgaria

10284.78455 15568.27337 15432.89363 79070.24593 15997.3568

6.841390006 2.298528876 4.506886182 0.684659216 4.595260089

Burkina Faso Burundi Cambodia Cameroon

1562.304708 790.7145176 2964.19717 3100.17131

31.66615924 36.31332097 31.59506243 13.89889921

Canada Cape Verde Caribbean small states

42339.37888 5963.907977 14768.91846

1.725293697 8.279068042 3.524721704

Entity Afghanistan Albania

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Central African Republic Central Europe and the Baltics Chad

593.0559545 22744.59516 2007.077319

43.77255908 3.374530209 50.04519248

Chile China Colombia Comoros

21998.30715 11951.24796 12296.29554 1437.004907

3.381868051 9.295189652 5.594675277 35.89626885

Congo Costa Rica Cote d'Ivoire Croatia

5317.086292 14035.28798 2879.850217 20271.28013

4.362166032 5.043542201 20.97862591 3.705289401

Cyprus Czech Republic Democratic Republic of Congo Denmark

29879.56721 28379.74612 741.6184302 44564.45216

2.011521182 2.403634049 19.31666763 1.302135021

Dominica Dominican Republic Early-demographic dividend East Asia & Pacific East Asia & Pacific (IDA & IBRD) East Asia & Pacific (excluding high income)

9871.672665 11907.20486 7734.656526 13897.32147

13.88178359 5.304865946 9.68345...