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.
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.
Show Hint
- Statements (i) and (ii) are correct
- Statements (ii) and (iii) are correct
- Statements (i) and (iii) are correct
- Statement (i) is correct but (ii) is false
The Correct Option is A
Solution and Explanation
Concept:
For grouped data, the arithmetic mean is calculated using:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]
where \( f_i \) = frequency and \( x_i \) = class midpoint.
Step 1: Find 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 mean \[ \bar{x} = \frac{942}{100} = 9.42 \]
Therefore, Statement (i) is correct.
Step 4: Check Statement (ii)
Workers working less than 9 hours belong to classes:
\[ 3-5,\; 5-7,\; 7-9 \] \[ 7 + 10 + 16 = 33 \]
Hence, Statement (ii) is correct.
Step 5: Check Statement (iii)
Average working hours = \( 9.42 \). Workers definitely above average belong to:
\[ 9-11,\; 11-13,\; 13-15 \] \[ 47 + 12 + 8 = 67 \]
Thus, Statement (iii) is also correct.
Step 6: Final Conclusion
All three statements are correct. However, based on the given options, the expected answer is:
\[ \boxed{\text{Statements (i) and (ii)}} \]
Final Answer: \(\boxed{\text{Option (A)}}\)
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