The problem provides a table showing the distribution of job seekers by gender and age group in a locality. We are asked to answer several questions about this data, including identifying the population and statistical unit, describing the observed character, constructing a histogram and frequency polygon, drawing a cumulative frequency curve, determining the modal class, median, and quartiles, and calculating the mean, interquartile range, interquartile interval, variance, and coefficient of variation for both men and women.

1. Problem Description

The problem provides a table showing the distribution of job seekers by gender and age group in a locality. We are asked to answer several questions about this data, including identifying the population and statistical unit, describing the observed character, constructing a histogram and frequency polygon, drawing a cumulative frequency curve, determining the modal class, median, and quartiles, and calculating the mean, interquartile range, interquartile interval, variance, and coefficient of variation for both men and women.

2. Solution Steps

1. Population and Statistical Unit:

The population studied is job seekers in a locality. The statistical unit is each individual job seeker.

2. Observed Character:

The observed characters are gender (qualitative) and age (quantitative continuous). The age is grouped into classes.

3. Histogram and Frequency Polygon:

To construct the histogram, the x-axis will represent the age groups (classes) and the y-axis will represent the frequencies (number of job seekers). We need to draw two separate histograms, one for men and one for women.

For the frequency polygon, we connect the midpoints of the tops of the bars in the histogram with straight lines. The frequency polygon must also start and end on the x-axis.

4. Cumulative Frequency Curve:

We need to calculate the cumulative frequencies for both men and women.

For example, for men, the cumulative frequencies are 280, 280+310=590, 590+240=830, 830+420=1250, 1250+70=

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0. For women, the cumulative frequencies are 160, 160+360=520, 520+120=640, 640+530=1170, 1170+50=

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0. Then, plot the points (upper limit of each class, cumulative frequency) for each gender and connect them with a smooth curve.

5. Modal Class, Median, and Quartiles:

a) Modal Class:

For men, the modal class is [50; 60[ because it has the highest frequency (420).

For women, the modal class is [50; 60[ because it has the highest frequency (530).

b) Median:

Total number of men is $280 + 310 + 240 + 420 + 70 = 1320$. The median is the value at position $1320/2 = 660$. The median class for men is [40; 50[.

Total number of women is $160 + 360 + 120 + 530 + 50 = 1220$. The median is the value at position $1220/2 = 610$. The median class for women is [40; 50[.

We can use linear interpolation to estimate the median.

For men: $Median = L + \frac{N/2 - CF}{f} * w$, where L is the lower limit of the median class (40), N is the total frequency (1320), CF is the cumulative frequency of the class before the median class (590), f is the frequency of the median class (240), and w is the class width (10).

$Median_{men} = 40 + \frac{660 - 590}{240} * 10 = 40 + \frac{70}{240} * 10 = 40 + 2.92 = 42.92$

For women: $Median = L + \frac{N/2 - CF}{f} * w$, where L is the lower limit of the median class (40), N is the total frequency (1220), CF is the cumulative frequency of the class before the median class (520), f is the frequency of the median class (120), and w is the class width (10).

$Median_{women} = 40 + \frac{610 - 520}{120} * 10 = 40 + \frac{90}{120} * 10 = 40 + 7.5 = 47.5$

c) Quartiles Q1 and Q3:

For men, Q1 is at position $1320/4 = 330$. The Q1 class for men is [26; 40[.

$Q1_{men} = 26 + \frac{330 - 280}{310} * 14 = 26 + \frac{50}{310} * 14 = 26 + 2.26 = 28.26$

For men, Q3 is at position $3*1320/4 = 990$. The Q3 class for men is [50; 60[.

$Q3_{men} = 50 + \frac{990 - 830}{420} * 10 = 50 + \frac{160}{420} * 10 = 50 + 3.81 = 53.81$

For women, Q1 is at position $1220/4 = 305$. The Q1 class for women is [26; 40[.

$Q1_{women} = 26 + \frac{305 - 160}{360} * 14 = 26 + \frac{145}{360} * 14 = 26 + 5.64 = 31.64$

For women, Q3 is at position $3*1220/4 = 915$. The Q3 class for women is [50; 60[.

$Q3_{women} = 50 + \frac{915 - 640}{530} * 10 = 50 + \frac{275}{530} * 10 = 50 + 5.19 = 55.19$

6. Mean, Interquartile Range, Interquartile Interval, Variance, and Coefficient of Variation:

We will approximate the mean by using the midpoint of each class. The midpoints are 21, 33, 45, 55, 62.

5. For men:

$Mean_{men} = (280*21 + 310*33 + 240*45 + 420*55 + 70*62.5) / 1320 = (5880 + 10230 + 10800 + 23100 + 4375) / 1320 = 54385 / 1320 = 41.2$

For women:

$Mean_{women} = (160*21 + 360*33 + 120*45 + 530*55 + 50*62.5) / 1220 = (3360 + 11880 + 5400 + 29150 + 3125) / 1220 = 52915 / 1220 = 43.37$

Interquartile Range:

$IQR = Q3 - Q1$

$IQR_{men} = 53.81 - 28.26 = 25.55$

$IQR_{women} = 55.19 - 31.64 = 23.55$

Interquartile Interval:

$[Q1, Q3]$

$[28.26, 53.81]$ for men

$[31.64, 55.19]$ for women

Variance:

$Var_{men} = \frac{\sum f_i(x_i - \bar{x})^2}{N} = \frac{280(21-41.2)^2 + 310(33-41.2)^2 + 240(45-41.2)^2 + 420(55-41.2)^2 + 70(62.5-41.2)^2}{1320} = \frac{280(408.04)+310(67.24)+240(14.44)+420(190.44)+70(453.69)}{1320}=\frac{114251.2+20844.4+3465.6+80000.8+31758.3}{1320}=\frac{250320.3}{1320}=189.64$

$Var_{women} = \frac{\sum f_i(x_i - \bar{x})^2}{N} = \frac{160(21-43.37)^2 + 360(33-43.37)^2 + 120(45-43.37)^2 + 530(55-43.37)^2 + 50(62.5-43.37)^2}{1220}=\frac{160(500.3969)+360(107.5369)+120(2.6569)+530(135.2169)+50(365.0569)}{1220}=\frac{80063.5+38713.3+318.8+71664.9+18252.8}{1220}=\frac{209013.3}{1220}=171.32$

Standard Deviation:

$SD = \sqrt{Variance}$

$SD_{men} = \sqrt{189.64} = 13.77$

$SD_{women} = \sqrt{171.32} = 13.1$

Coefficient of Variation:

$CV = \frac{SD}{Mean}$

$CV_{men} = \frac{13.77}{41.2} = 0.334$

$CV_{women} = \frac{13.1}{43.37} = 0.302$

3. Final Answer

1. The population studied is job seekers in a locality. The statistical unit is each individual job seeker.

2. The observed characters are gender (qualitative) and age (quantitative continuous).

3. Histograms and frequency polygons (separately) should be constructed as described above.

4. Cumulative frequency curves (separately) should be constructed as described above.

5. Modal class: Men [50; 60[, Women [50; 60[; Median: Men 42.92, Women 47.5; Q1: Men 28.26, Women 31.64; Q3: Men 53.81, Women 55.

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6. Mean: Men 41.2, Women 43.37; Interquartile Range: Men 25.55, Women 23.55; Interquartile Interval: Men [28.26, 53.81], Women [31.64, 55.19]; Variance: Men 189.64, Women 171.32; Coefficient of Variation: Men 0.334, Women 0.

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