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Aurora Textile Company

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ASE 2009

Case Study

Record 40 – 60 data for two statistical variables (X and Y) at your choice.

I. For each of the two variables:

a) Calculate and interpret the average, standard deviation and the coefficient of variation for row data. Interpret the results. Is the data series homogenous?
b) Summarize the data in an appropriate number of classes. Construct the frequency distribution.
c) Calculate and interpret for the frequency distribution the average, standard deviation and coefficient of variance. Compare with the results from point a). Explain the differences.
d) Construct a histogram and describe the shape of the distribution based on the histogram.
e) In which interval is expected that about 95% of the data will fall? Is this assumption true for this data?

II. Using the “Pivot Table Wizard” in EXCEL, build a pivot table on your spreadsheet (using also the second variable). You may have to change the order of the rows (You should define the intervals first using VLookup function.

III. Calculate the regression line and interpret and test the regression coefficients, coefficient of determination and coefficient of correlation. Interpret the results.


We chose for analysis a sample of 41 countries, considering the relationship between the following two variables: the Human Development Index(HDI) and the Gross Product per Capita (GDP)[1]. The values are extracted from a report provided by the World Bank in 2008.

The gross domestic product (GDP) or gross domestic income (GDI) is a basic measure of a country's economic performance and is the market value of all final goods and services made within the borders of a country in a year. It is a fundamental measurement of production and is very often positively correlated with the standard of living, though its use as a stand-in for measuring progress in increasing the standard of living has come under increasing criticism and many countries are actively exploring alternative measures. GDP can be defined in three ways, all of which are conceptually identical. First, it is equal to the total expenditures for all final goods and services produced within the country in a stipulated period of time (usually a 365-day year). Second, it is equal to the sum of the value added at every stage of production (the intermediate stages) by all the industries within a country, plus taxes less subsidies on products, in the period. Third, it is equal to the sum of the income generated by production in the country in the period—that is, compensation of employees, taxes on production and imports less subsidies, and gross operating surplus (or profits).

The Human Development Index (HDI) is an index used to rank countries by level of "human development", which usually also implies whether a country is developed, developing, or underdeveloped. The HDI combines normalized measures of life expectancy, educational attainment, and GDP per capita for countries worldwide. It is claimed as a standard means of measuring human development—a concept that, according to the United Nations Development Program (UNDP), refers to the process of widening the options of persons, giving them greater opportunities for education, health care, income, employment, etc. The HDI attempts to measure a country's development.

The HDI combines three dimensions:

• Life expectancy at birth, as an index of population health and longevity • Knowledge and education, as measured by the adult literacy rate (with two-thirds weighting) and the combined primary, secondary, and tertiary gross enrollment ratio (with one-third weighting). • Standard of living, as measured by the natural logarithm of gross domestic product per capita at purchasing power parity.

I. a)

| |HDI |GDP per capita |
|average |0,93414634 |44448 |
|standard deviation |0,0384113 |13170,96478 |
|coeficient of variation |0,04111915 |0,296320237 |

Interpretation for HDI

The average of the HDI data is 0.934146. Looking at the values from HDI column, we can see that there are few extreme values, therefore the average did not suffer any distortions. The coefficient of variation is 4% (less than 30%), so the series of data collected is homogeneous. Regarding the standard deviation, this is 3.8, so the HDI individual values deviate with +/- 3.8 units from the average.

Interpretation for GDP

The computed average for the GDP data is 44448. The coefficient of variation is 29.63% , which is smaller than 30%, showing that the data series is homogeneous. Regarding the standard deviation, this is 13170.96478, so the GDP deviates with +/- 13170 from the average.

I. b)

In order to group the data into classes, the range was first calculated. For the GDP values, the range was 53735 units, with a minimum of 20945 and a maximum of 74680. Having in mind that for a set of data with less than 50 recordings, the number of classes should be between 5 and 7, we considered to group the data into six classes, each class having a range of 10000 units.

Frequency distribution:

|LL (GDP) |UL (GDP) |intervals |No. of |class midpoints(xi) |xi*fi |(xi-avg)^2*fi |
| | | |Countries(fi) | | | |
|30000 |40000 |30000-40000 |11 |35000 |385000 |1423296875 |
|40000 |50000 |40000-50000 |12 |45000 |540000 |22687500 |
|50000 |60000 |50000-60000 |7 |55000 |385000 |520734375 |
|60000 |70000 |60000-70000 |3 |65000 |195000 |1040671875 |
|70000 |80000 |70000-80000 |3 |75000 |225000 |2458171875 |
|TOTAL | | |41 |300000 |1855000 |7750015625 |

For the HDI values, the range was 0.165, with a minimum of 0.806 and a maximum of 0.971. Thus, we grouped this series of data into 5 classes, each class having a range of 0.4 units.

Frequency distribution:

|LL (HDI) |UL (HDI) |Intervals |No. of Countries(fi) |class midpoints(xi) |xi*fi |(xi-avg)^2*fi |
|0,84 |0,88 |0.84-0.88 |1 |0,86 |0,86 |0,00178042 |
|0,88 |0,92 |0.88-0.92 |9 |0,9 |8,1 |524,313569 |
|0,92 |0,96 |0.92-0.96 |21 |0,94 |19,74 |7931,96767 |
|0,96 |1 |0.96-1 |8 |0,98 |7,84 |389,170311 |
| Average |46375 |
|Standard deviation |13748,62798 |
|Coefficient of variance |23764166666,67 |

|Average |0,9545 |
|Standard deviation |14,6896728 |
|Coefficient of variance |57,15748 |

|Results from point 1a. |HDI |GDP per Capita |
|Average |0,934146341 |44448 |
|Standard deviation |0,038411301 |13170,96478 |
|Coefficient of variation |0,041119147 |0,296320237 |

|Average |1927 |0,02035366 |
|Standard deviation |577,66320360 |14,65126150 |

I. d) Histogram


The skewness for the GDP per Capita is 0.465491928 (positively skewed). Thus the distribution is skewed to the right.


The skewness for the HDI is -1.979992312 (negatively skewed). Thus the distribution is skewed to the left.

I. e)

Gross Domestic Product per Capita

Formula: [pic];

Confidence interval:


[pic]44028.63, 44867.37];

Hypothesis testing

I. H0: µ=44471 H1: µ≠44471

II. [pic]

III. -0.00027>-1.96 => accept H0 => the assumption that 95% of the data will fall in the interval [44028.63, 44867.37] is true.

-1.96 1.96 - 0.00027

Human Development Index

Formula: [pic];

Confidence interval:


[pic]0.816567, 1.051678573];

Hypothesis testing

I. H0: µ=0.96 H1: µ≠0.96

II. [pic]

III. -0.0105>-1.96 => accept H0 => the assumption that 95% of the data will fall in the interval [0.816567, 1.051678573] is true.

-1.96 1.96


II. Using the Vlookup function the GDP and also HDI had been sorted in 3 categories, based on their dimension[3]. For example as shown in the data array for the GDP we considered that a GDP between 20000 and 40000 is considered a small GDP per capita. As for HDI, the values are considered to be small if found between 0.8 and 0.87. Using EXCEL’s PivotTable Wizard, we obtain a table which highlights the relationship between our two variables. Based on the above mentioned table a large GDP per capita is associated with 5 large and 1 medium value of HDI, and so on.

Data array GDP

|LL |interval | |
|20000 |20000-40000 |small |
|40000 |40000-60000 |medium |
|60000 |60000-80000 |large |

Data array HDI

|LL |interval | |
|0.8 |0.8-0.87 |small |
|0.87 |0.87-0.94 |medium |
|0.94 |0.94-1 |large |

Pivot table

HDI count of vlookup

| | |
| | |
| | |
| |large |medium |small |Grand Total |
|large |5 |1 | |6 |
|medium |15 |4 | |19 |
|small |5 |8 |3 |16 |
|Grand Total |25 |13 |3 |41 |



Regression equation: y = 0.00000169x + 0.85902992

The slope is 0.00000169, which means that for each increase of one unit in X, the Y is estimated to increase by 0.00000169 units. For each increase of 1 unit in the GDP per capita, the model predicts that the HDI is estimated to increase by 0.00000169 units.

When x=0, y=0.85902992. This means the trend line reaches OY axes in the value 0.85902992.
The intercept is a positive number, thus the regression line is ascending.

Regression coefficients testing:

We chose 2 values from our database to test the regression line:

| HDI | GDP per capita |
|0.806 |20945 |
|0.814 |25095 |

y = 0.00000169*20945 + 0.85902992≈0.806

y = 0.00000169*25095 + 0.85902992≈0.814

The P value in the ANOVA table[4] has a value under 1% (1.47535801187148E-36

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