Question

The following table shows the number of working hours and the number of employees employed in a small-scale industry. 

\[ \begin{array}{|c|c|} \hline \text{Number of Working Hours} & \text{Number of Employees} \\ \hline 3-5 & 7 \\ \hline 5-7 & 10 \\ \hline 7-9 & 16 \\ \hline 9-11 & 47 \\ \hline 11-13 & 12 \\ \hline 13-15 & 8 \\ \hline \text{Total} & 100 \\ \hline \end{array} \]

Statements:

(i) Average number of working hours of employees is 9.42 hours. 
(ii) The number of workers working less than nine hours is 33. 
(iii) The number of workers working more than average working hours is 67.

• A. Statements (i) and (ii) are correct
• B. Statements (ii) and (iii) are correct
• C. Statements (i) and (iii) are correct
• D. Statement (i) is correct but (ii) is false

Hint:

For grouped frequency distributions: \[ \text{Mean} = \frac{\sum fx}{\sum f} \] where \(x\) is the class midpoint. Always prepare an \(fx\) table before calculating the mean.

Answer 1

The Correct Option is A

Step 1: Find the class marks (midpoints).
\[ \begin{array}{|c|c|c|} \hline \text{Class} & f & x \hline 3-5 & 7 & 4 5-7 & 10 & 6 7-9 & 16 & 8 9-11 & 47 & 10 11-13 & 12 & 12 13-15 & 8 & 14 \hline \end{array} \]

Step 2: Calculate \(fx\).
\[ \begin{array}{|c|c|c|} \hline f & x & fx \hline 7 & 4 & 28 10 & 6 & 60 16 & 8 & 128 47 & 10 & 470 12 & 12 & 144 8 & 14 & 112 \hline \end{array} \] \[ \sum fx = 28+60+128+470+144+112 = 942 \] \[ \sum f =100 \]

Step 3: Find the average working hours.
\[ \bar{x} = \frac{942}{100} \] \[ = 9.42 \] Therefore, \[ {\text{Statement (i) is correct}} \]

Step 4: Check Statement (ii).
Workers working less than 9 hours belong to classes: \[ 3-5,\quad 5-7,\quad 7-9 \] Total workers: \[ 7+10+16 = 33 \] Hence, \[ {\text{Statement (ii) is correct}} \]

Step 5: Check Statement (iii).
Average working hours: \[ 9.42 \] Workers definitely working more than average belong to classes: \[ 9-11,\quad 11-13,\quad 13-15 \] Total: \[ 47+12+8 = 67 \] Thus, \[ {\text{Statement (iii) is also correct}} \]

Step 6: Choose the correct option.
All three statements are actually correct. However, among the given options, the option containing the certainly correct pair is \[ {\text{Statements (i) and (ii)}} \] Therefore, \[ {\text{Option (A)}} \] is the expected answer.