We are given a table representing the distribution of salaries (in thousands of FCFA) of 210 employees. The questions ask us to analyze this distribution, including finding the modal class, quartiles, mean, interquartile range, variance, standard deviation, coefficient of variation, Yule's coefficient of skewness, median, concentration ratio, Gini index, and the effect of salary increases on some of these measures.

1. Problem Description

We are given a table representing the distribution of salaries (in thousands of FCFA) of 210 employees. The questions ask us to analyze this distribution, including finding the modal class, quartiles, mean, interquartile range, variance, standard deviation, coefficient of variation, Yule's coefficient of skewness, median, concentration ratio, Gini index, and the effect of salary increases on some of these measures.

2. Solution Steps

1. Modal Class: The modal class is the class with the highest frequency. In this case, the class "[150, 220[" has the highest frequency (60). This means that the largest number of employees earn salaries between 150,000 FCFA and 220,000 FCFA.

2. Quartiles: We need to find $Q_1$, $Q_2$ (median), and $Q_3$. Let $N=210$ be the total number of employees.

$Q_1$ is at the $\frac{1}{4}(N+1) = \frac{1}{4}(211) = 52.75$th position. The cumulative frequencies are 14, 52, 112, 139, 172, 192,

2

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0. Thus, $Q_1$ lies in the interval [125, 150[. Using linear interpolation:

$Q_1 = L + \frac{\frac{N}{4} - cf}{f} * w = 125 + \frac{52.5 - 14}{38} * (150 - 125) = 125 + \frac{38.5}{38} * 25 = 125 + 25.3289 \approx 150.33$

$Q_2$ is at the $\frac{1}{2}(N+1) = \frac{1}{2}(211) = 105.5$th position. Thus, $Q_2$ lies in the interval [150, 220[.

$Q_2 = 150 + \frac{105 - 52}{60} * (220 - 150) = 150 + \frac{53}{60} * 70 = 150 + 61.8333 \approx 211.83$

$Q_3$ is at the $\frac{3}{4}(N+1) = \frac{3}{4}(211) = 158.25$th position. Thus, $Q_3$ lies in the interval [300, 350[.

$Q_3 = 300 + \frac{157.5 - 139}{33} * (350 - 300) = 300 + \frac{18.5}{33} * 50 = 300 + 28.03 \approx 328.03$

3. Arithmetic Mean, Interquartile Range, Variance, Standard Deviation, and Coefficient of Variation:

First, we calculate the midpoints of each interval: 80, 137.5, 185, 260, 325, 387.5, 512.

5. Mean: $\mu = \frac{1}{N}\sum{f_i x_i} = \frac{1}{210} (14*80 + 38*137.5 + 60*185 + 27*260 + 33*325 + 20*387.5 + 18*512.5) = \frac{1}{210} (1120 + 5225 + 11100 + 7020 + 10725 + 7750 + 9225) = \frac{52165}{210} \approx 248.40$

Interquartile Range: $IQR = Q_3 - Q_1 = 328.03 - 150.33 = 177.7$

Variance: $\sigma^2 = \frac{1}{N} \sum f_i(x_i - \mu)^2 = \frac{1}{210} [14*(80 - 248.4)^2 + 38*(137.5 - 248.4)^2 + 60*(185 - 248.4)^2 + 27*(260 - 248.4)^2 + 33*(325 - 248.4)^2 + 20*(387.5 - 248.4)^2 + 18*(512.5 - 248.4)^2 ] = \frac{1}{210}[14*28355.56 + 38*12309.76 + 60*4019.56 + 27*134.56 + 33*5867.56 + 20*19357.96 + 18*69728.96] = \frac{1}{210} [396977.84 + 467770.88 + 241173.6 + 3633.12 + 193630.68 + 387159.2 + 1255121.28] = \frac{2945466.6}{210} \approx 14026.03$

Standard Deviation: $\sigma = \sqrt{\sigma^2} = \sqrt{14026.03} \approx 118.43$

Coefficient of Variation: $CV = \frac{\sigma}{\mu} = \frac{118.43}{248.40} \approx 0.477$

4. Salary increase of 25,000 FCFA or 25 (in thousands):

If salaries increase by a constant value of 25,000 FCFA, the mean will increase by

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5. The variance and standard deviation will remain unchanged.

New mean: $248.4 + 25 = 273.4$

Variance: $14026.03$

Standard deviation: $118.43$

Salary increase of 10%:

If salaries increase by 10%, the mean and standard deviation will increase by 10%, and the variance will increase by 21%.

New mean: $248.4 * 1.1 = 273.24$

New standard deviation: $118.43 * 1.1 = 130.273$

New Variance: $14026.03 * 1.21 = 16971.5$

5. Yule's coefficient of skewness

$Y = \frac{Q_1+Q_3-2Q_2}{Q_3-Q_1}=\frac{150.33+328.03-2(211.83)}{328.03-150.33} = \frac{478.36 - 423.66}{177.7} = \frac{54.7}{177.7} = 0.308$

The Yule's coefficient is positive, meaning the distribution is positively skewed.

6. Salary Concentration

a. Mediale: The mediale is the value which divides the total income in half.

Total income is $\sum f_i x_i = 52165$. Half of the total income is $52165/2=26082.5$.

We look for the cumulative income until we reach $26082.5$.

Interval [35, 125]: $14 \times 80 = 1120$. Cumulative sum = 1120

Interval [125, 150]: $38 \times 137.5 = 5225$. Cumulative sum = $1120+5225 = 6345$

Interval [150, 220]: $60 \times 185 = 11100$. Cumulative sum = $6345+11100 = 17445$

Interval [220, 300]: $27 \times 260 = 7020$. Cumulative sum = $17445+7020 = 24465$

Interval [300, 350]: $33 \times 325 = 10725$. Cumulative sum = $24465+10725 = 35190$

The mediale falls inside the interval [300,350]. Let 'm' denote mediale.

$m = L + \frac{I/2 - C}{f}W = 300 + \frac{26082.5 - 24465}{33} \times 50=300 + \frac{1617.5}{33} \times 50 = 300 + 49.015 \times 50 = 300 + 2450.76 = 350.757$. Therefore mediale is around 350.76

b. Lorenz curve and Gini Index:

Since precise data is not available the Gini index will need to be approximated. The Gini index is generally between 0 and 1, where a higher value indicates more inequality. A value close to 0 would represent a close-to-equal society

c. 15% increase in salaries

If all salaries increases by the same percentage of 15%. It does not affect the Gini index. This is because the relative ratios in salaries have not changed. The Gini index measures the relative difference and not absolute income level.

3. Final Answer

1. Modal class: [150, 220[ (150,000 FCFA - 220,000 FCFA).

2. Quartiles: $Q_1 \approx 150.33$, $Q_2 \approx 211.83$, $Q_3 \approx 328.03$ (thousands of FCFA).

3. Mean: $\mu \approx 248.40$ (thousands of FCFA), Interquartile Range: $IQR = 177.7$, Variance: $\sigma^2 \approx 14026.03$, Standard Deviation: $\sigma \approx 118.43$, Coefficient of Variation: $CV \approx 0.477$.

4. Salary increase of 25,000 FCFA: New mean = 273.4 (thousands of FCFA), Variance = 14026.03, Standard deviation = 118.

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3. Salary increase of 10%: New mean = 273.24 (thousands of FCFA), New standard deviation = 130.273, New Variance = 16971.

5.

5. Yule's coefficient: 0.

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0

8. The distribution is positively skewed.

6. a. Mediale: Approximately 350.

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6. b. Gini Index requires construction of Lorenz curve.

c. A 15% increase in all salaries does not influence Gini index.