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.