The problem provides a frequency distribution table of the annual local taxes paid by the residents of a municipality, in thousands of CFA Francs. Several tasks are to be performed based on this data, including identifying the population and variable, constructing histograms and cumulative frequency curves, calculating descriptive statistics such as the mode, median, quartiles, mean, interquartile range, variance, and coefficient of variation. Further analysis involves studying the asymmetry of the frequency polygon using Yule's coefficient of skewness and analyzing the concentration of local taxes using the concentration curve of Gini and the Gini index.

1. Problem Description

The problem provides a frequency distribution table of the annual local taxes paid by the residents of a municipality, in thousands of CFA Francs. Several tasks are to be performed based on this data, including identifying the population and variable, constructing histograms and cumulative frequency curves, calculating descriptive statistics such as the mode, median, quartiles, mean, interquartile range, variance, and coefficient of variation. Further analysis involves studying the asymmetry of the frequency polygon using Yule's coefficient of skewness and analyzing the concentration of local taxes using the concentration curve of Gini and the Gini index.

2. Solution Steps

1. Population and Variable:

* The population studied is the residents of the municipality.

* The observed variable is the annual amount of local taxes paid (in thousands of CFA Francs).

* The nature of the variable is quantitative and continuous.

2. Descriptive Statistics Calculation:

First, calculate the total number of observations (N).

$N = 1 + 7 + 11 + 8 + 12 + 15 + 19 + 16 + 8 + 3 = 100$

(a) Modal Class: The modal class is the class with the highest frequency. The modal class is [12,16[ with a frequency of

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9.

(b) Median: The median is the value that divides the dataset into two equal halves. To find the median class, find the class that contains the $N/2 = 100/2 = 50$th observation.

Cumulative frequencies: 1, 8, 19, 27, 39, 54, 73, 89, 97,

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0

0. The median class is [10, 12[.

To calculate the median, we use the formula:

$Median = L + \frac{\frac{N}{2} - cf}{f} * w$

where $L$ is the lower limit of the median class (10), $N$ is the total number of observations (100), $cf$ is the cumulative frequency of the class before the median class (39), $f$ is the frequency of the median class (15), and $w$ is the class width (2).

$Median = 10 + \frac{50 - 39}{15} * 2 = 10 + \frac{11}{15} * 2 = 10 + 1.4667 = 11.4667$

(c) Quartiles:

* Q1 (First Quartile): Q1 is the value that separates the lowest 25% of the data. Find the class that contains the $N/4 = 100/4 = 25$th observation.

The Q1 class is [8,9[.

$Q1 = L + \frac{\frac{N}{4} - cf}{f} * w$

$Q1 = 8 + \frac{25 - 19}{8} * 1 = 8 + \frac{6}{8} = 8.75$

* Q3 (Third Quartile): Q3 is the value that separates the lowest 75% of the data. Find the class that contains the $3N/4 = 300/4 = 75$th observation.

The Q3 class is [16, 20[.

$Q3 = L + \frac{\frac{3N}{4} - cf}{f} * w$

$Q3 = 16 + \frac{75 - 73}{16} * 4 = 16 + \frac{2}{16} * 4 = 16 + 0.5 = 16.5$

(d) Arithmetic Mean: To calculate the arithmetic mean, we first need to find the midpoint of each class: 3, 5, 7, 8.5, 9.5, 11, 14, 18, 30,

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0. $Mean = \frac{\sum (midpoint * frequency)}{N}$

$Mean = \frac{(3*1) + (5*7) + (7*11) + (8.5*8) + (9.5*12) + (11*15) + (14*19) + (18*16) + (30*8) + (60*3)}{100}$

$Mean = \frac{3 + 35 + 77 + 68 + 114 + 165 + 266 + 288 + 240 + 180}{100} = \frac{1436}{100} = 14.36$

(e) Interquartile Range: $IQR = Q3 - Q1 = 16.5 - 8.75 = 7.75$

Semi-Interquartile Range: $SIQR = \frac{Q3 - Q1}{2} = \frac{7.75}{2} = 3.875$

(f) Variance:

$Variance = \frac{\sum f_i(x_i - \bar{x})^2}{N}$, where $f_i$ is the frequency, $x_i$ is the midpoint of each class and $\bar{x}$ is the mean.

Calculating each term:

(3 - 14.36)^2 * 1 = 129.0496

(5 - 14.36)^2 * 7 = 607.1824

(7 - 14.36)^2 * 11 = 599.9864

(8.5 - 14.36)^2 * 8 = 272.9024

(9.5 - 14.36)^2 * 12 = 276.1416

(11 - 14.36)^2 * 15 = 168.786

(14 - 14.36)^2 * 19 = 2.3744

(18 - 14.36)^2 * 16 = 209.0176

(30 - 14.36)^2 * 8 = 1869.4208

(60 - 14.36)^2 * 3 = 6262.6224

Total = 10427.4736

$Variance = \frac{10427.4736}{100} = 104.274736$

(g) Coefficient of Variation:

$CV = \frac{\sqrt{Variance}}{Mean} = \frac{\sqrt{104.274736}}{14.36} = \frac{10.2115}{14.36} = 0.7109$

3. Yule's Coefficient of Skewness:

a. Domain of Variation of Yule's coefficient of skewness $CD = \frac{Q3+Q1-2Median}{Q3-Q1}$ is [-1, 1].

b. If $CD = 0$, it means the distribution is symmetrical, and the median is the average of the first and third quartiles: $Q1 + Q3 = 2 * Median$.

c. Calculate $CD$:

$CD = \frac{16.5+8.75-2*11.4667}{16.5-8.75} = \frac{25.25 - 22.9334}{7.75} = \frac{2.3166}{7.75} = 0.2989$

Since $CD > 0$, the distribution is positively skewed (skewed to the right).

4. Concentration of local taxes.

a. The concentration range is the difference between the highest and lowest amount of tax paid. Concentration range = $80 - 2 = 78$ (in thousands of Francs CFA).

3. Final Answer

1. Population studied: Residents of the municipality.

Observed variable: Annual amount of local taxes paid (in thousands of CFA Francs).

Nature of the variable: Quantitative and continuous.

2. Descriptive Statistics:

Modal Class: [12,16[

Median: 11.4667

Q1: 8.75

Q3: 16.5

Mean: 14.36

Interquartile Range: 7.75

Variance: 104.274736

Coefficient of Variation: 0.7109

3. Yule's Coefficient of Skewness:

a. Domain: [-1, 1]

b. $CD = 0$ signifies a symmetrical distribution.

c. $CD = 0.2989$. The distribution is positively skewed.

4. Concentration of local taxes

a. Concentration range: 78