What is the Relationship between the Gross Domestic Product (GDP) and the CD sales Rate of a Country?

Math Studies Internal Assessment

International School Bangkok

Ayaka Nishimura P.1

Teacher: Mr. De Mille

Word Count:

What is the Relationship between the Gross Domestic Product (GDP) and

the CD Sales Rate of a Country?

Introductio n

Statement of Task

The main purpose of this investigation is to determine whether there is a relationship

between a countries’ gross domestic product and its CD sales rate . The Gross Domestic Product

(GDP) is the total value of all goods and services produced in a country, in one year. The CD

sales rate of the country is the number of the CDs that the country sold in a year.

Plan of Investigation

Following the data collection, a number of mathematical processes were used to analyze

the data; standard deviation, least square regression, Pearson’s correlation coefficient and the

Chi-square test.

Collected Data

Table 1: GDP and CD Sales Rates for 25 Countries (2005)

# Country GDP CD Sales (million)1 USA 11.7 300.52 Japan 3.745 93.73 UK 1.782 66.84 Germany 2.362 58.75 France 1.737 47.36 Italy 1.609 14.77 Canada 1.023 20.88 Australia 0.6117 14.59 India 3.319 10.910 Spain 0.9376 17.511 Netherlands 0.481 8.712 Russia 1,408 25.513 Mexico 1,006 33.414 Brazil 1,492 17.615 Austria 0.2559 4.516 Switzerland 0.2519 7.117 Belgium 0.3162 6.718 Norway 0.183 4.519 Sweden 0.2554 6.620 Denmark 0.1744 4

Table 1: Table 1 displays the data that was collected from CIA World Fact book 2007 for the GDP, and IFPI organization for CD sales.

Data Analysis / Mathematical Processes

Graph 1 shows the GDP vs. CD sales platted on a scatter plot. As of now, it appears as if it is a positive correlation which is moderate.

Standard Deviation Calculations

Standard Deviation measures the variability/dispersion of the particular variables (in this case, of HDI and divorce rates). It is given by the following formulae:

2.494 is the standard deviation of x, the GDP rates. This indicates a condensed range of

data yet sufficient for a statistical analysis.

0.88 is the standard deviation of y, the CD sales rates. Similar to the GDP rates’

standard deviation, this indicates a condensed range of data .

Least Squares Regression

Least Square regression calculation identify the relationship between the

independent variable, x, and the dependent variable, y. The least square

regression is given by the following formulae:

where =

Therefore:

y = 24.7097 – 81.0097 is the least squares regression formula for this particular set of

data. As can be seen later on in Graph 2, this is corresponds to the calculation made by

Microsoft Excel.

Pearson’s Correlation Coefficient

Pearson's correlation coefficient indicates the strength of the relationship between the

two variables (GDP and CD sales rate). It is given by the following formula:

The following is the previously displayed Laggor Pro graph with the program’s

calculation of the least squares regression line.

Graph 2 indicates that there is a moderate positive linear correlation. This is also

indicated through the value of the correlation coefficient, 0.9625. (Graph and

calculation were generated by Laggor Pro).

Chi-Square Test

Chi-square test measures the independence of the two variables. The following formulas

are used:

Observed Values:

B1 B2 Total

A1 a b a+b

A2 c d c +d

Total a+c b+d N

Calculations of Expected Values:

B1 B2 Total

A1 a+b

A2 c+d

Total a+c b+d N

χ2 ＝

Degree of freedom measures the number of values in the calculation that can vary:

Df ＝(r－1)(c－1)

r; row, c; column

Null (Ho) Hypothesis: HDI and divorce rates are independent

Alternative (Hl) Hypothesis: HDI and divorce rates are not independent.

Table 2: Observation Values

0-19.9 20-39.9 40-59.9 60-79.9 80-99.9 Total

3+ 1 0 0 0 1 2

2-2.9 0 0 1 0 0 1

1-1.9 2 3 1 1 0 7

0-0.9 9 0 0 0 0 9

Total 12 3 2 1 1 19

Table 2 shows the observed values for GDP vs. CD sales rates. The data of America

is excepted because it is an outlier.

Table 3: Calculations for the Expected Values

0-19.9 20-39.9 40-59.9 60-79.9 80-99.9 Total

3+ 2

2-2.9 1

1-1.9 7

Gro

ss D

om

estic

Pro

duct

(GD

P)

Rate

CD Sales Rate

CD Sales Rate

Gro

ss D

om

estic

Pro

duct

(GD

P)

Rate

0-0.9 9

Total 12 3 2 1 1 19

Table 3 shows the individual calculations for each of the expected

values.

Table 4: Expected Values

0-19.9 20-39.9 40-59.9 60-79.9 80-99.9 Total

3+ 1.263 0.316 0.211 0.105 0.105 2

2-2.9 0.632 0.158 0.105 0.053 0.053 1

1-1.9 4.421 0.105 0.737 0.368 0.368 7

0-0.9 5.684 1.421 0.947 0.474 0.474 9

Total 12 3 2 1 1 19

Table 4 shows the expected values, retrieved by the calculations in table

2.

CD Sales RateGro

ss D

om

estic

Pro

duct

(GD

P)

Rate

Df = (4-1) (5-1) 12

Discussion

Data Interpretation

In the beginning, the Graph 2 indicates that there is a moderate positive linear

correlation. This is also indicated through the value of the correlation coefficient, 0.9625

which was generated along with the graph in Laggor Pro. Furthermore, it is also

supported by Pearson’s correlation coefficient, which in its turn shows as 0.90779.

Therefore, all these calculations somewhat support the notion that the gross domestic

product (GDP) rate of the country does have an effect on the CD sales rate.

Finally, the chi-square test demonstrates the as the critical value is less than the

chi square value, thus the null hypothesis is rejected and the alternative hypothesis,

which states that the GDP and the CD sales are dependent, is embraced.

These two results show that CD sales in countries somewhat influence on

countries’ GDP. It is considered that countries with higher GDP are developed countries

with strong economic power, so consumers in such developed countries have stronger

purchasing power.

The critical value at 5% significance with 12 degrees of freedom is 21.0261. As the chi square value (28.006) > than the critical value, the null hypothesis is embraced and we can conclude that HDI and divorce rates are not independent.

Limitation

One large limitation of the data collected was that most of data are from

developed countries, so the result that there is high correlation between GDP and CD

sales are of course reasonable since there is not much difference between countries the

number of CD a person purchase. Therefore, there should be data of developing

countries.

Another major limitation is that since the price of a CD is not expensive,

people in developing countries can also buy it. So it is not sure if it supports the

purchasing power.

Conclusion

In spite of the previously mentioned limitations, the investigation still points

toward the fact that there is a moderate positive linear correlation between GDP and CD

sales rates. Furthermore, the investigation does hold a sense of validity and the obtained

result is very reasonable. However, there is a defect if one sees it by the viewpoint

“comparing countries’ economic power and purchasing power” since there is no data of

developing country included.