We are given a frequency distribution of the number of cars observed at a toll booth during 5-minute intervals, based on 100 observations. We need to answer the following questions: 1. Identify the characteristic being studied, its nature, and its modalities.

1. Problem Description

We are given a frequency distribution of the number of cars observed at a toll booth during 5-minute intervals, based on 100 observations. We need to answer the following questions:

1. Identify the characteristic being studied, its nature, and its modalities.

2. Represent the distribution graphically using a bar chart, a frequency polygon, and a cumulative frequency diagram.

3. Determine the mode, median, and quartiles $Q1$ and $Q3$. Interpret the results.

4. Calculate the arithmetic mean, geometric mean, harmonic mean, quadratic mean, variance, standard deviation, and coefficient of variation for this series.

2. Solution Steps

1. (a) The characteristic being studied is the number of cars presenting themselves at the toll booth during a 5-minute interval.

(b) The nature of the characteristic is quantitative discrete (since the number of cars can only be an integer value).

(c) The modalities of the characteristic are the observed values of the number of cars: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,

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2.

2. Graphical representations:

- Bar chart: A bar chart with the number of cars on the x-axis (1 to 12) and the number of observations on the y-axis (2, 8, 14, 20, 19, 15, 9, 6, 2, 3, 1, 1).

- Frequency polygon: Plot the points with coordinates (number of cars, number of observations) as (1, 2), (2, 8), (3, 14), (4, 20), (5, 19), (6, 15), (7, 9), (8, 6), (9, 2), (10, 3), (11, 1), (12, 1) and connect them with lines.

- Cumulative frequency diagram:

Calculate the cumulative frequencies: 2, 10, 24, 44, 63, 78, 87, 93, 95, 98, 99,

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0. Plot the points with coordinates (number of cars, cumulative frequency): (1, 2), (2, 10), (3, 24), (4, 44), (5, 63), (6, 78), (7, 87), (8, 93), (9, 95), (10, 98), (11, 99), (12, 100). Connect these points.

3. (a) Mode: The mode is the value with the highest frequency. In this case, the highest frequency is 20, corresponding to 4 cars. So, the mode is

4. (b) Median: The median is the middle value when the data is ordered. Since there are 100 observations, the median is the average of the 50th and 51st values. The cumulative frequencies tell us that the 44th observation is 4 cars, and the 63rd observation is 5 cars. Therefore, the 50th and 51st values are

5. So, the median is

5. (c) Quartile Q1: $Q1$ is the value such that 25% of the observations are below it. 25% of 100 is

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5. The cumulative frequency 24 corresponds to 3 cars, and the cumulative frequency 44 corresponds to 4 cars. Therefore, $Q1$ is

4. (d) Quartile Q3: $Q3$ is the value such that 75% of the observations are below it. 75% of 100 is

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5. The cumulative frequency 63 corresponds to 5 cars, and the cumulative frequency 78 corresponds to 6 cars. Therefore, $Q3$ is

6.

Interpretation:

The most frequent number of cars observed in a 5-minute interval is

4. Half of the observations had 5 or fewer cars. 25% of the observations had 4 or fewer cars, and 75% of the observations had 6 or fewer cars.

4. (a) Arithmetic mean $(\bar{x})$:

$\bar{x} = \frac{\sum x_i f_i}{\sum f_i}$

$\bar{x} = \frac{(1 \times 2) + (2 \times 8) + (3 \times 14) + (4 \times 20) + (5 \times 19) + (6 \times 15) + (7 \times 9) + (8 \times 6) + (9 \times 2) + (10 \times 3) + (11 \times 1) + (12 \times 1)}{100}$

$\bar{x} = \frac{2 + 16 + 42 + 80 + 95 + 90 + 63 + 48 + 18 + 30 + 11 + 12}{100} = \frac{507}{100} = 5.07$

(b) Geometric mean $(G)$:

$G = \sqrt[N]{\prod x_i^{f_i}}$

$G = \sqrt[100]{1^2 \times 2^8 \times 3^{14} \times 4^{20} \times 5^{19} \times 6^{15} \times 7^9 \times 8^6 \times 9^2 \times 10^3 \times 11^1 \times 12^1}$

Taking the logarithm:

$\ln(G) = \frac{1}{100} [2\ln(1) + 8\ln(2) + 14\ln(3) + 20\ln(4) + 19\ln(5) + 15\ln(6) + 9\ln(7) + 6\ln(8) + 2\ln(9) + 3\ln(10) + \ln(11) + \ln(12)]$

$\ln(G) = \frac{1}{100} [0 + 8(0.693) + 14(1.099) + 20(1.386) + 19(1.609) + 15(1.792) + 9(1.946) + 6(2.079) + 2(2.197) + 3(2.303) + 2.398 + 2.485]$

$\ln(G) = \frac{1}{100} [5.544 + 15.386 + 27.720 + 30.571 + 26.880 + 17.514 + 12.474 + 4.394 + 6.909 + 2.398 + 2.485] = \frac{152.275}{100} = 1.52275$

$G = e^{1.52275} \approx 4.583$

(c) Harmonic mean $(H)$:

$H = \frac{N}{\sum \frac{f_i}{x_i}}$

$H = \frac{100}{\frac{2}{1} + \frac{8}{2} + \frac{14}{3} + \frac{20}{4} + \frac{19}{5} + \frac{15}{6} + \frac{9}{7} + \frac{6}{8} + \frac{2}{9} + \frac{3}{10} + \frac{1}{11} + \frac{1}{12}}$

$H = \frac{100}{2 + 4 + 4.667 + 5 + 3.8 + 2.5 + 1.286 + 0.75 + 0.222 + 0.3 + 0.091 + 0.083} = \frac{100}{24.649} \approx 4.057$

(d) Quadratic mean $(Q)$:

$Q = \sqrt{\frac{\sum x_i^2 f_i}{\sum f_i}}$

$Q = \sqrt{\frac{(1^2 \times 2) + (2^2 \times 8) + (3^2 \times 14) + (4^2 \times 20) + (5^2 \times 19) + (6^2 \times 15) + (7^2 \times 9) + (8^2 \times 6) + (9^2 \times 2) + (10^2 \times 3) + (11^2 \times 1) + (12^2 \times 1)}{100}}$

$Q = \sqrt{\frac{2 + 32 + 126 + 320 + 475 + 540 + 441 + 384 + 162 + 300 + 121 + 144}{100}} = \sqrt{\frac{3047}{100}} = \sqrt{30.47} \approx 5.520$

(e) Variance $(\sigma^2)$:

$\sigma^2 = \frac{\sum (x_i - \bar{x})^2 f_i}{\sum f_i} = \frac{\sum x_i^2 f_i}{\sum f_i} - \bar{x}^2$

$\sigma^2 = Q^2 - \bar{x}^2 = 30.47 - (5.07)^2 = 30.47 - 25.7049 = 4.7651$

(f) Standard deviation $(\sigma)$:

$\sigma = \sqrt{\sigma^2} = \sqrt{4.7651} \approx 2.183$

(g) Coefficient of variation $(CV)$:

$CV = \frac{\sigma}{\bar{x}} \times 100 = \frac{2.183}{5.07} \times 100 \approx 43.06\%$

3. Final Answer

1. (a) Characteristic: Number of cars. (b) Nature: Quantitative discrete. (c) Modalities: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,

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2. See the explanation in the solution steps.

3. Mode: 4, Median: 5, Q1: 4, Q3:

6. Interpretation: The most frequent number of cars observed in a 5-minute interval is

4. Half of the observations had 5 or fewer cars. 25% of the observations had 4 or fewer cars, and 75% of the observations had 6 or fewer cars.

4. Arithmetic mean: 5.07, Geometric mean: 4.583, Harmonic mean: 4.057, Quadratic mean: 5.520, Variance: 4.7651, Standard deviation: 2.183, Coefficient of variation: 43.06%.