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.

The total number of employees is $\sum f = 20 + 35 + 30 + 10 + 5 = 100$.

The estimated mean travelling time is $\boxed{\text{19.5 minutes}}$.

2.2 Write the modal class of the data.

From the table, the highest frequency is 35, which corresponds to the class interval $10 \le x < 20$.
The modal class is $\boxed{\text{10 \le x < 20}}$.

2.3 Construct a box-whisker plot for the data in the ANSWER BOOK.

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 ($3N/4 = 3 \times 100/4 = 75$). 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.

We calculated the mean $\bar{x} = 19.5$ 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 $\boxed{\text{TRUE}}$.

2.4.2 The inter-quartile range for the data is 25.

IQR is calculated as $Q3 - Q1$.
Using the values from the ogive: $Q3 = 28$ minutes and $Q1 = 7$ minutes.

Since the calculated IQR is 21 minutes, not 25 minutes, the statement is $\boxed{\text{FALSE}}$.

2.4.3 35 employees 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 $0 \le x < 10$ and $10 \le x < 20$.
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 $\boxed{\text{FALSE}}$.