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As a piece of Statistics coursework, I have decided to compare two items of data, in order to prove, or disprove my theory:

Extracts from this document...

Introduction

Grant Mackenzie 11F

Statistics Coursework

As a piece of Statistics coursework, I have decided to compare two items of data, in order to prove, or disprove my theory: “A country’s position in the Commonwealth games varies accordingly to that country’s population size.”

My theory is that a country’s position in something such as the Olympics or Commonwealth Games is proportional to that country’s population size. I say this because I believe that if a country has a large population, there will be more potential athletes to choose from.

I am doing this because I would be genuinely interested in finding out whether or not this theory is true, and I believe that it is a theory that many people reading this essay would be curious in finding out. In addition, I am comparing the results from the Commonwealth Games, instead of something as renowned as the Olympic games because the Commonwealth Games are dominated by countries with very different traditions and cultures. Conversely, countries that dominate the Olympic games are countries such as France, England, or Germany, and are all countries that live a very western way of life similar to ours.    

In order to do the comparisons that I will need to make properly, I will use three different occasions of the Commonwealth Games, and then make an average for the number of medals awarded for each country. I will use the most recent of games – 2002, 1998 and 1994. I will do this, as the data collected from these three games, will contain the data from each of the countries that enter, since in the first games in 1930, only a fraction of the countries that enter now entered.

...read more.

Middle

Sri Lanka

19,576,783

St Helena

7,367

St Kitts

43054

St Lucia

150,157

St Vincent & The Grenadines

120,519

Swaziland

1,123,605

Tanzania

37,187,939

Tonga

106,137

Trinidad & Tobago

1,163,724

Turks & Calcos

18,738

Tuvalu

11,146

Uganda

24,699,073

Vanuatu

196,178

Wales

2,903,085

Zambia

9,959,037

Zimbabwe

11,376,676

Now that I have both the data for the countries population sizes and the amounts of medals awarded to them, I can test my theory in a pilot test.

 I will select ten different samples from the finite population that I collected using stratified random sampling. By saying random, I mean that the out coming country cannot be predicted and is chosen without conscious decision.

There are many types of sampling that can be done including simple random sampling, stratified sampling, systematic sampling, cluster sampling, quota sampling, convenience sampling and opinion polls.

Simple Random Sampling – In this type of sampling, every sample unit within the population has an equal chance of being chosen.

Stratified Sampling – For this type of sampling, the population is divided into strata (categories) and then a random sample is chosen from each of the strata within the population. The size of each sample is in proportion to the size of each stratum (category) within the population.

Systematic Sampling – As the name suggests, systematic sampling is where a regular pattern is devised to choose the sample. Every item in the population is listed and a starting point is chosen at random, with every nth item being selected.

Cluster Sampling – For cluster sampling, the population is divided into groups, or like the name suggested clusters. A random sample of groups or clusters is chosen and every item in the chosen cluster is surveyed.

Quota Sampling – In quota sampling, instructions are given concerning the amount (quota) of each section of the population to be sampled.

Convenience Sampling – Convenience sampling is by far the easiest sampling to make as it is, as the name suggests, convenient. The most convenient sample is chosen which for thirty countries, could be the first thirty countries in the list.

Opinion Polls – Opinion polls, as the name suggests are large-scale opinion polls that often use a combination of cluster and quota sampling.

As you can see, stratified sampling is the most suited type of sampling that I can use. Stratification of sampling is necessary when the sampling frame is significantly non-homogeneous (which tends to be true of most human populations and I believe is true of this exercise). Some characteristics will be shared but most will be influenced by cultural, socio-economic, gender, religious and ethnic differences. For example, I believe that countries in the developing world (e.g. Zimbabwe, Malaysia), who do not have the benefit of the intense training that athletes in the developed world (e.g. England, Australia) endure, will not win as many medals.

Firstly, I will categorise each the population into stratum. I will do this by using the method in which the Commonwealth Games’ website used – separating each country into that country’s locality (Asia, Oceania, Europe, Caribbean and the Americas).

Below are the strata that I have made:

LOCALITY

COUNTRY

POPULATION (2002)

Americas

Belize

262,999

Bermuda

63,960

Canada

31,902,268

Falkland Islands

2,967

Guyana

698,209

St Helena

7,367

Africa

Botswana

1,591,232

Cameroon

16,184,748

Gambia

1,455,842

Ghana

20,244,154

Kenya

31,138,735

Lesotho

2,207,954

Malawi

10,701,824

Mauritius

1,200,206

Mozambique

19,607,519

Namibia

1,820,916

Nigeria

129,934,911

Seychelles

80,098

Sierra Leone

5,614,743

South Africa

43,647,658

Swaziland

1,123,605

Uganda

24,699,073

Tanzania

37,187,939

Zambia

9,959,037

Zimbabwe

11,376,676

Asia

Bangladesh

133,376,684

Brunei

350,898

India

1,045,845,226

Malaysia

22,662,365

Maldives

320,165

Pakistan

147,663,429

Singapore

4,452,732

Sri Lanka

19,576,783

Caribbean

Anguilla

12,446

Antigua and Barbuda

67,448

Bahamas

300,529

Barbados

276,607

British Virgin Islands

21,272

Cayman Islands

36,273

Dominica

70,158

Grenada

89,211

Jamaica

2,680,029

Montserrat

8,437

St Kitts

43,054

St Lucia

150,157

St Vincent & The Grenadines

120,519

Trinidad & Tobago

1,163,724

Turks & Calcos

18,738

Europe

England

49,138,831

Cyprus

767,314

Gibraltar

27,714

Guernsey

64,587

Isle of Man

76,535

Jersey

89,775

Malta

397,499

Northern Ireland

1,685,267

Scotland

5,062,011

Wales

2,903,085

Oceania

Australia

19,546,792

Cook Islands

20,811

Fiji

856,346

Kiribati

96,335

Nauru

12,329

New Zealand

3,908,037

Niue

2,134

Norfolk Islands

1,866

Papua New Guinea

5,172,033

Samoa

178,631

Solomon Islands

494,786

Tonga

106,137

Tuvalu

11,146

Vanuata

196,178

Now that I have made the strata, I can now take samples from them. Before I can do this, I must determine the number of samples that will be chosen from each stratum -

Samples to be taken from the Americas  =∑ (American Strata)   x 50

                                                                      ∑ (Population)

        =6   x 50

            72

        =0.083… x 50

        =4.16

        =4

To avoid any confusion, I ‘truncated’ the answer, which means that I simply ‘cut off’ the decimal. This will negate any rounding errors that may occur. I will do this for each of the other strata.

Now that I have determined the number of samples to be chosen from the Americas, I must use a way to decide which sample will be chosen. In order to do this, I will use the ‘random number generator’ on my calculator. The random number generator chooses, at random, a decimal number. Then I must simply multiply that number by the number of sample units that there are within that stratum. Since if you were to get a small decimal, when you multiplied it by any integer, the answer will be less that 1 so therefore, I will add 1 to the answer. Seeing that the random number generator will only give me a number, I will have to label each of the sample units within the strata. Below are the listings that I have created for the Americas –

LOCALITY

COUNTRY

POPULATION (2002)

NUMBER LISTING

Americas

Belize

262,999

1

Bermuda

63,960

2

Canada

31,902,268

3

Falkland Islands

2,967

4

Guyana

698,209

5

St Helena

7,367

6

Now that I have created a listing for each sample unit, I will use the random number generator in order to pick a unit at random. Now I must do this four different times.

1st sample unit to be chosen  =   (Random # x 6) + 1

       =   (0.81 x 6) + 1

                                                      =4.86 + 1

                                                      =5.86

                                                      =5

 2nd sample unit to be chosen  =   (Random # x 6) + 1

         =   (0.112 x 6) + 1

                                                        =0.672 + 1

                                                        =1.672

                                                        =1

3rd sample unit to be chosen  =   (Random # x 6) + 1

       =   (0.381 x 6) + 1

                                                      =2.286 + 1

                                                      =3.286

                                                      =3

4th sample unit to be chosen  =   (Random # x 6) + 1

       =   (0.785 x 6) + 1

                                                      =4.71+ 1

                                                      =5.71

                                                      =5

As you can see, 5 has appeared once already, so I must try again –

4th sample unit to be chosen  =   (Random # x 6) + 1

        =   (0.638 x 6) + 1

                                                       =3.828 + 1

                                                       =4.428

                                                       =4

Now that I have chosen the sample units to be chosen, which are 5,1,3,4, I can now translate those numbers to the countries, Guyana, Belize, Canada and Falkland Islands.

        Since I have chosen the sample units that will be chosen from the Americas, I will now choose the sample units for Africa. But first, I will need to create a listing for Africa, much like I did for the Americas –

LOCALITY

COUNTRY

POPULATION (2002)

NUMBER LISTING

Africa

Botswana

1,591,232

1

Cameroon

16,184,748

2

Gambia

1,455,842

3

Ghana

20,244,154

4

Kenya

31,138,735

5

Lesotho

2,207,954

6

Malawi

10,701,824

7

Mauritius

1,200,206

8

Mozambique

19,607,519

9

Namibia

1,820,916

10

Nigeria

129,934,911

11

Seychelles

80,098

12

Sierra Leone

5,614,743

13

South Africa

43,647,658

14

Swaziland

1,123,605

15

Uganda

24,699,073

16

Tanzania

37,187,939

17

Zambia

9,959,037

18

Zimbabwe

11,376,676

19

Now I will need to find out how many samples will need to be taken from this stratum –

Samples to be taken from the Africa  =∑ (Africa Strata)   x 50

                                                               ∑ (Population)

        =19   x 50

             72

        =0.2638… x 50

        =13.19

        =13

Therefore, I will need to choose 13 different samples from the stratum –

1st sample unit to be chosen  =   (Random # x 19) + 1

       =   13.319 + 1

                                                      =14.319

                                                      =14

2nd sample unit to be chosen  =   (Random # x 19) + 1

       =   1.387 + 1

                                                      =2.387

                                                      =2

3rd sample unit to be chosen  =   (Random # x 19) + 1

       =   18.131 + 1

                                                      =19.131

                                                      =19

4th sample unit to be chosen  =   (Random # x 19) + 1

       =   10.051 + 1

                                                      =11.051

                                                      =11

5th sample unit to be chosen  =   (Random # x 19) + 1

       =   8.018 + 1

                                                      =9.018

                                                      =9

6th sample unit to be chosen  =   (Random # x 19) + 1

       =   15.884 + 1

                                                      =16.884

                                                      =16

7th sample unit to be chosen  =   (Random # x 19) + 1

       =   16.562 + 1

                                                      =17.562

                                                      =17

8th sample unit to be chosen  =   (Random # x 19) + 1

       =   0.114 + 1

                                                      =1.114

                                                      =1

9th sample unit to be chosen  =   (Random # x 19) + 1

       =   2.66 + 1

                                                      =3.66

                                                      =3

10th sample unit to be chosen  =   (Random # x 19) + 1

        =   4.218 + 1

                                                       =5.218

                                                       =5

11th sample unit to be chosen  =   (Random # x 19) + 1

        =   11.628 + 1

                                                       =12.628

                                                       =12

12th sample unit to be chosen  =   (Random # x 19) + 1

        =   17.043 + 1

                                                       =18.043

                                                       =18

13th sample unit to be chosen  =   (Random # x 19) + 1

        =   12.065 + 1

                                                       =13.065

                                                       =13

As you can see, I have chosen the four different sample units to be chosen, which translates to the countries, South Africa, Cameroon, Tanzania, Nigeria, Mozambique, Uganda, Zimbabwe, Botswana, Gambia, Kenya, Seychelles, Zambia and Sierra Leone.

        Now, like Africa and the Americas, I will label the different countries for Asia –

LOCALITY

COUNTRY

POPULATION (2002)

NUMBER LISTING

Asia

Bangladesh

133,376,684

1

Brunei

350,898

2

India

1,045,845,226

3

Malaysia

22,662,365

4

Maldives

320,165

5

Pakistan

147,663,429

6

Singapore

4,452,732

7

Sri Lanka

19,576,783

8

...read more.

Conclusion

I must now repeat the procedure once more for the region of Oceania –  

LOCALITY

COUNTRY

POPULATION (2002)

NUMBER LISTING

Oceania

Australia

19,546,792

1

Cook Islands

20,811

2

Fiji

856,346

3

Kiribati

96,335

4

Nauru

12,329

5

New Zealand

3,908,037

6

Niue

2,134

7

Norfolk Islands

1,866

8

Papua New Guinea

5,172,033

9

Samoa

178,631

10

Solomon Islands

494,786

11

Tonga

106,137

12

Tuvalu

11,146

13

Vanuata

196,178

14

Samples to be taken from Oceania  =∑ (Oceania Strata)   x 50

                                                             ∑ (Population)

 =14   x 50

      72

 =0.194… x 50

 =9.722…

                                                               =9

1st sample unit to be chosen  =   (Random # x 14) + 1

       =   1.064 + 1

                                                      =2.064

                                                      =2

2nd sample unit to be chosen  =   (Random # x 14) + 1

       =   7.938 + 1

                                                      =8.938

                                                      =8

3rd sample unit to be chosen  =   (Random # x 14) + 1

       =   6.384 + 1

                                                      =7.384

                                                      =7

4th sample unit to be chosen  =   (Random # x 14) + 1

       =   13.636 + 1

                                                      =14.636

                                                      =14

5th sample unit to be chosen  =   (Random # x 14) + 1

       =   5.936 + 1

                                                      =6.936

                                                      =6

6th sample unit to be chosen  =   (Random # x 14) + 1

       =   10.64 + 1

                                                      =11.64

                                                      =11

7th sample unit to be chosen  =   (Random # x 14) + 1

       =   4.86 + 1

                                                      =5.86

                                                      =5

8th sample unit to be chosen  =   (Random # x 14) + 1

       =   12.838 + 1

                                                      =13.838

                                                      =13

9th sample unit to be chosen  =   (Random # x 14) + 1

       =   2.428 + 1

                                                      =3.428

                                                      =3

These numbers translate to the countries, the Cook Islands, Norfolk Islands, Niue, Vanuata, New Zealand, the Solomon Islands, Nauru, Tuvalu and Fiji.

I have now collected all of the samples that I will be using. Below is the complete listing of the countries that I have chosen through the random number generator and through stratified sampling –

REGION

COUNTRY

Americas

Guyana

Belize

Canada

Falkland Islands

Africa

South Africa

Cameroon

Zimbabwe

Nigeria

Mozambique

Uganda

Tanzania

Botswana

Gambia

Kenya

Seychelles

Zambia

Sierra Leone

Asia

Brunei

India

Sri Lanka

Maldives

Bangladesh

Caribbean

Cayman Islands

St Kitts

Montserrat

St Lucia

Barbados

Jamaica

Anguilla

Grenada

Turks & Calcos

Trinidad & Tobago

Europe

Wales

Malta

Isle of Man

Gibralta

Jersey

Northern Ireland

Oceana

Cook Islands

Norfolk Islands

Niue

Vanuata

New Zealand

Solomon Islands

Nauru

Tuvalu

Fiji

NOTE: There are 47 countries here instead of the originally intended 50. This is because of errors in truncating the number of samples to be taken.

Now I shall compare each country’s population with their amount of total number of medals awarded –

Country

Average No. of Medals Awarded

Population Size

Guyana

0

698,209

Belize

0

262,999

Canada

109

31,902,268

Falkland Islands

0

2,967

South Africa

38

43,647,658

Cameroon

8

16,184,748

Zimbabwe

4

11,376,676

Nigeria

13

129,934,911

Mozambique

1

19,607,519

Uganda

2

24,699,073

Tanzania

2

37,187,939

Botswana

1

1,591,232

Gambia

0

1,455,842

Kenya

16

31,138,735

Seychelles

1

80,098

Zambia

2

9,959,037

Sierra Leone

0

5,614,743

Brunei

0

350,898

India

46

1,045,845,226

Sri Lanka

1

19,576,783

Maldives

0

320,165

Bangladesh

0

133,376,684

Cayman Islands

0

36,273

St Kitts

0

43,054

Montserrat

0

8,437

St Lucia

0

150,157

Barbados

1

276,607

Jamaica

11

2,680,029

Anguilla

0

12,446

Grenada

0

89,211

Turks & Calcos

0

18,738

Trinidad & Tobago

2

1,163,724

Wales

21

2,903,085

Malta

0

397,499

Isle of Man

0

76,535

Gibraltar

0

27,714

Jersey

0

89,775

Northern Ireland

6

1,685,267

Cook Islands

0

20,811

Norfolk Islands

0

1,866

Niue

0

2,134

Vanuata

0

196,178

New Zealand

40

3,908,037

Solomon Islands

0

494,786

Nauru

8

12,329

Tuvalu

0

11,146

Fiji

2

856,346

 Now I must compare this data. In order to this, I will do a scatter graph.

...read more.

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Related AS and A Level Probability & Statistics essays

  1. GCSE Mathematics Coursework: Statistics Project

    Otherwise, the proportions of girls and boys in the sample would not represent the proportions present in the Year Group. Also, as the random numbers generated can be up to three decimal places, they will be rounded to the nearest whole number.

  2. Study of the height/diameter ratio of limpets inhabiting the middle shore region of exposed ...

    This happened on the sheltered shore more than the other two; the distributions obtained could be different from the ones in reality; the conclusions reached could therefore be imprecise. Orientation of limpets is difficult to take into account because one has to assume where the front of the limpet lies without being able to actually know.

  1. The aim of this investigation was to look at the reliability and validity of ...

    They are essentially intended as research tools (as opposed to diagnostic tools for use in clinical settings) and, as such, 'they are regarded as acceptable, reliable and valid' (Kline 1981, Shackleton and Fletcher, 1984). We also need to ensure that the test measures what it is supposed to measure, bringing in the question of validity.

  2. Statistics. I have been asked to construct an assignment regarding statistics. The statistics ...

    Finding the Mean, Median and Mode In order to find the Mean, I will add all of the attendance numbers together, and divide the number of sets of data I recorded; Birmingham City Mean: 21,394 + 27,333 + 22,186 + 23,138 + 26,850 + 26,474 + 24,357 + 25,770 +

  1. "The lengths of lines are easier to guess than angles. Also, that year 11's ...

    For the year 11's the inter-quartile range was 9� and the median was 43�, this means that the year 11's had a lower inter-quartile range value, but the year 9's had a lower median value. After drawing cumulative frequency curves, I was able to draw some box plots.

  2. Statistics coursework

    This is necessary to see whereabouts the majority of results lie. After this I have chosen to produce a stem and leaf diagram of girls and boys IQ. This is because a stem and leaf diagram has the same advantages as a bar chart i.e.

  1. Anthropometric Data

    This is like an inverse correlation which shows the direction of the correlation slopping down. No correlation When observing the pattern on this particular scatter it shows that there were no relationships the different variables. Dependent and Independent variables Dependent and independent variables gives the understanding to refer values that change in the relationship to each other.

  2. Teenagers and Computers Data And Statistics Project

    Number of red faces Formula 0 (n - 2 ) 3 1 6 [ (n - 2) 2] 2 n3 - (n -2)3 - 6[(n - 2 )2 ] - 8 3 8 Total n 3 5. Explanation of the 10 x 10 x 10 cube This was relatively easy

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