The problem presents a table showing the distribution of annual profits (in millions of CFA francs) of 50 companies in the Dakar region. We are asked to identify the population and statistical unit, the observed variable and its nature, to calculate the centered moments of order 2, 3, and 4, and to determine Fisher's asymmetry coefficient and Pearson's kurtosis coefficient. Finally, we must interpret the calculated values.

1. Problem Description

The problem presents a table showing the distribution of annual profits (in millions of CFA francs) of 50 companies in the Dakar region. We are asked to identify the population and statistical unit, the observed variable and its nature, to calculate the centered moments of order 2, 3, and 4, and to determine Fisher's asymmetry coefficient and Pearson's kurtosis coefficient. Finally, we must interpret the calculated values.

2. Solution Steps

1. Population and Statistical Unit:

* The population studied is the set of 50 companies in the Dakar region.

* The statistical unit is a single company in the Dakar region.

2. Observed Variable and its Nature:

* The observed variable is the annual profit of the companies.

* The nature of this variable is quantitative continuous since the profit can take any value within a certain range.

3. Calculating Centered Moments:

First, we need to calculate the mean. We will approximate each interval by its midpoint.

The midpoints are: 7.5, 32.5, 62.5, and

1

0

0.

Mean $(\mu) = \frac{\sum x_i f_i}{\sum f_i} = \frac{(7.5 \times 15) + (32.5 \times 17) + (62.5 \times 10) + (100 \times 8)}{50} = \frac{112.5 + 552.5 + 625 + 800}{50} = \frac{2090}{50} = 41.8$

Now we calculate the centered moments.

* Centered moment of order 2 (Variance):

$\mu_2 = \frac{\sum (x_i - \mu)^2 f_i}{\sum f_i} = \frac{(7.5-41.8)^2 \times 15 + (32.5-41.8)^2 \times 17 + (62.5-41.8)^2 \times 10 + (100-41.8)^2 \times 8}{50} = \frac{(-34.3)^2 \times 15 + (-9.3)^2 \times 17 + (20.7)^2 \times 10 + (58.2)^2 \times 8}{50} = \frac{1176.49 \times 15 + 86.49 \times 17 + 428.49 \times 10 + 3387.24 \times 8}{50} = \frac{17647.35 + 1470.33 + 4284.9 + 27097.92}{50} = \frac{50500.5}{50} = 1010.01$

* Centered moment of order 3:

$\mu_3 = \frac{\sum (x_i - \mu)^3 f_i}{\sum f_i} = \frac{(7.5-41.8)^3 \times 15 + (32.5-41.8)^3 \times 17 + (62.5-41.8)^3 \times 10 + (100-41.8)^3 \times 8}{50} = \frac{(-34.3)^3 \times 15 + (-9.3)^3 \times 17 + (20.7)^3 \times 10 + (58.2)^3 \times 8}{50} = \frac{-40423.807 \times 15 + (-804.357) \times 17 + 8869.743 \times 10 + 196807.848 \times 8}{50} = \frac{-606357.105 - 13674.069 + 88697.43 + 1574462.784}{50} = \frac{1003129.04}{50} = 20062.5808$

* Centered moment of order 4:

$\mu_4 = \frac{\sum (x_i - \mu)^4 f_i}{\sum f_i} = \frac{(7.5-41.8)^4 \times 15 + (32.5-41.8)^4 \times 17 + (62.5-41.8)^4 \times 10 + (100-41.8)^4 \times 8}{50} = \frac{(-34.3)^4 \times 15 + (-9.3)^4 \times 17 + (20.7)^4 \times 10 + (58.2)^4 \times 8}{50} = \frac{1390925.8881 \times 15 + 7130.7121 \times 17 + 183052.6321 \times 10 + 11451775.7056 \times 8}{50} = \frac{20863888.3215 + 121222.1057 + 1830526.321 + 91614205.6448}{50} = \frac{114431542.393}{50} = 2288630.84786$

4. Fisher's Asymmetry Coefficient and Pearson's Kurtosis Coefficient:

* Fisher's Asymmetry Coefficient $(\gamma_1)$:

$\gamma_1 = \frac{\mu_3}{\sigma^3} = \frac{\mu_3}{(\mu_2)^{3/2}}$

Since $\mu_2 = 1010.01$, then $\sigma = \sqrt{1010.01} = 31.78$

$\gamma_1 = \frac{20062.5808}{(31.78)^3} = \frac{20062.5808}{32055.34} = 0.626$

* Pearson's Kurtosis Coefficient $(\gamma_2)$:

$\gamma_2 = \frac{\mu_4}{\sigma^4} - 3 = \frac{\mu_4}{(\mu_2)^2} - 3$

$\gamma_2 = \frac{2288630.84786}{(1010.01)^2} - 3 = \frac{2288630.84786}{1020120.1}-3 = 2.2435 - 3 = -0.7565$

Interpretation:

* Fisher's asymmetry coefficient ($\gamma_1$ = 0.626) is positive, indicating that the distribution is skewed to the right (positively skewed). This means there are more companies with profits below the mean than above it, and the tail is longer on the right side.

* Pearson's kurtosis coefficient ($\gamma_2$ = -0.7565) is negative, indicating that the distribution is platykurtic. This means that the distribution is flatter and has thinner tails compared to a normal distribution.

3. Final Answer

1. Population: 50 companies in the Dakar region. Statistical Unit: A single company in the Dakar region.

2. Observed variable: Annual profit of the companies. Nature: Quantitative continuous.

3. Centered moments: $\mu_2 = 1010.01$, $\mu_3 = 20062.5808$, $\mu_4 = 2288630.84786$

4. Fisher's Asymmetry Coefficient: 0.

6

2

6. Pearson's Kurtosis Coefficient: -0.

7

5

6

5. The distribution is positively skewed and platykurtic.