The following table gives the frequency distribution of the daily travelling time (in minutes) from home to work for the employees of a certain company.

Question

Asked on July 24, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-24T16:43:06.478Z

2.1 Calculate the estimated mean travelling time. Step 1: Calculate the midpoint (x_m) for each interval and the product of frequency (f) and midpoint (x_m). The total number of employees is f = 20 + 35 + 30 + 10 + 5 = 100. | Daily travelling time x (in minutes) | Number of employees (f) | Midpoint of Interval (x_m) | f · x_m | | :------------------------------------- | :------------------------ | :--------------------------- | :------------ | | 0 x < 10 | 20 | (0+10)/(2) = 5 | 20 × 5 = 100 | | 10 x < 20 | 35 | (10+20)/(2) = 15 | 35 × 15 = 525 | | 20 x < 30 | 30 | (20+30)/(2) = 25 | 30 × 25 = 750 | | 30 x < 40 | 10 | (30+40)/(2) = 35 | 10 × 35 = 350 | | 40 x < 50 | 5 | (40+50)/(2) = 45 | 5 × 45 = 225 | | Total | 100 | | 1950 | Step 2: Calculate the estimated mean using the formula x = ( (f · x_m))/( f). x = (1950)/(100) = 19.5 The estimated mean travelling time is 19.5 minutes. 2.2 Write the modal class of the data. Step 1: Identify the class interval with the highest frequency. From the table, the highest frequency is 35, which corresponds to the class interval 10 x < 20. The modal class is 10 x < 20. 2.3 Construct a box-whisker plot for the data in the ANSWER BOOK. Step 1: Determine the five-number summary from the given ogive (cumulative frequency curve). The total number of employees is N = 100. Minimum value: The lowest travelling time is 0 minutes. Maximum value: The highest travelling time is 50 minutes. Median (Q2): This is the value corresponding to the 50^th employee (N/2 = 100/2 = 50). Reading from the ogive at a cumulative frequency of 50, the travelling time is approximately 18 minutes. First Quartile (Q1): This is the value corresponding to the 25^th employee (N/4 = 100/4 = 25). Reading from the ogive at a cumulative frequency of 25, the travelling time is approximately 7 minutes. Third Quartile (Q3): This is the value corresponding to the 75^th employee (3N/4 = 3 × 100/4 = 75). Reading from the ogive at a cumulative frequency of 75, the travelling time is approximately 28 minutes. The five-number summary for the box-whisker plot is: Minimum = 0 minutes Q1 = 7 minutes Median = 18 minutes Q3 = 28 minutes Maximum = 50 minutes To construct the box-whisker plot: • Draw a number line that covers the range from 0 to 50 minutes. • Mark the minimum value (0) and the maximum value (50) with whiskers. • Draw a box from the first quartile (Q1 = 7) to the third quartile (Q3 = 28). • Draw a vertical line inside the box at the median (Q2 = 18). 2.4 State whether the following statements are TRUE or FALSE. 2.4.1 The distribution of these travelling times is positively skewed. Step 1: Compare the mean and median. We calculated the mean x = 19.5 minutes. We determined the median from the ogive as 18 minutes. Since the mean (19.5) is greater than the median (18), the distribution is positively skewed. Additionally, the frequency distribution shows a peak towards lower values and a longer tail towards higher values, which is characteristic of positive skewness. The statement is TRUE. 2.4.2 The inter-quartile range for the data is 25. Step 1: Calculate the inter-quartile range (IQR). IQR is calculated as Q3 - Q1. Using the values from the ogive: Q3 = 28 minutes and Q1 = 7 minutes. IQR = 28 - 7 = 21 minutes Since the calculated IQR is 21 minutes, not 25 minutes, the statement is FALSE. 2.4.3 35 employees take less than 20 minutes. Step 1: Determine the number of employees who take less than 20 minutes. From the frequency table, the number of employees taking less than 20 minutes is the sum of frequencies for the intervals 0 x < 10 and 10 x < 20. Number of employees = 20 + 35 = 55 employees. Alternatively, reading from the ogive at x=20 minutes, the cumulative frequency is 55. Since 55 employees take less than 20 minutes, not 35, the statement is FALSE. Drop the next question! 📸

Accepted Answer · 2026-07-24T16:43:06.478Z

2.1 Calculate the estimated mean travelling time. Step 1: Calculate the midpoint (x_m) for each interval and the product of frequency (f) and midpoint (x_m). The total number of employees is f = 20 + 35 + 30 + 10 + 5 = 100. | Daily travelling time x (in minutes) | Number of employees (f) | Midpoint of Interval (x_m) | f · x_m | | :------------------------------------- | :------------------------ | :--------------------------- | :------------ | | 0 x < 10 | 20 | (0+10)/(2) = 5 | 20 × 5 = 100 | | 10 x < 20 | 35 | (10+20)/(2) = 15 | 35 × 15 = 525 | | 20 x < 30 | 30 | (20+30)/(2) = 25 | 30 × 25 = 750 | | 30 x < 40 | 10 | (30+40)/(2) = 35 | 10 × 35 = 350 | | 40 x < 50 | 5 | (40+50)/(2) = 45 | 5 × 45 = 225 | | Total | 100 | | 1950 | Step 2: Calculate the estimated mean using the formula x = ( (f · x_m))/( f). x = (1950)/(100) = 19.5 The estimated mean travelling time is 19.5 minutes. 2.2 Write the modal class of the data. Step 1: Identify the class interval with the highest frequency. From the table, the highest frequency is 35, which corresponds to the class interval 10 x < 20. The modal class is 10 x < 20. 2.3 Construct a box-whisker plot for the data in the ANSWER BOOK. Step 1: Determine the five-number summary from the given ogive (cumulative frequency curve). The total number of employees is N = 100. Minimum value: The lowest travelling time is 0 minutes. Maximum value: The highest travelling time is 50 minutes. Median (Q2): This is the value corresponding to the 50^th employee (N/2 = 100/2 = 50). Reading from the ogive at a cumulative frequency of 50, the travelling time is approximately 18 minutes. First Quartile (Q1): This is the value corresponding to the 25^th employee (N/4 = 100/4 = 25). Reading from the ogive at a cumulative frequency of 25, the travelling time is approximately 7 minutes. Third Quartile (Q3): This is the value corresponding to the 75^th employee (3N/4 = 3 × 100/4 = 75). Reading from the ogive at a cumulative frequency of 75, the travelling time is approximately 28 minutes. The five-number summary for the box-whisker plot is: Minimum = 0 minutes Q1 = 7 minutes Median = 18 minutes Q3 = 28 minutes Maximum = 50 minutes To construct the box-whisker plot: • Draw a number line that covers the range from 0 to 50 minutes. • Mark the minimum value (0) and the maximum value (50) with whiskers. • Draw a box from the first quartile (Q1 = 7) to the third quartile (Q3 = 28). • Draw a vertical line inside the box at the median (Q2 = 18). 2.4 State whether the following statements are TRUE or FALSE. 2.4.1 The distribution of these travelling times is positively skewed. Step 1: Compare the mean and median. We calculated the mean x = 19.5 minutes. We determined the median from the ogive as 18 minutes. Since the mean (19.5) is greater than the median (18), the distribution is positively skewed. Additionally, the frequency distribution shows a peak towards lower values and a longer tail towards higher values, which is characteristic of positive skewness. The statement is TRUE. 2.4.2 The inter-quartile range for the data is 25. Step 1: Calculate the inter-quartile range (IQR). IQR is calculated as Q3 - Q1. Using the values from the ogive: Q3 = 28 minutes and Q1 = 7 minutes. IQR = 28 - 7 = 21 minutes Since the calculated IQR is 21 minutes, not 25 minutes, the statement is FALSE. 2.4.3 35 employees take less than 20 minutes. Step 1: Determine the number of employees who take less than 20 minutes. From the frequency table, the number of employees taking less than 20 minutes is the sum of frequencies for the intervals 0 x < 10 and 10 x < 20. Number of employees = 20 + 35 = 55 employees. Alternatively, reading from the ogive at x=20 minutes, the cumulative frequency is 55. Since 55 employees take less than 20 minutes, not 35, the statement is FALSE. Drop the next question! 📸

The following table gives the frequency distribution of the daily travelling time (in minutes) from home to work for the employees of a certain company.

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The following table gives the frequency distribution of the daily travelling time (in minutes) from home to work for the employees of a certain company.

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