The monthly salaries of the workers of a company are normally distributed with mean Rs. 24000 and standard deviation Rs. 4000. If 334 workers get a monthly salary of less than Rs. 18000, then the total number of workers in the company is (Given \(P(0 \le Z \le 1.5) = 0.4332\), where Z is the standard normal variate)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5000
Let X be the salary of a worker, so \(\mu = 24000\) and \(\sigma = 4000\). For X = 18000,
\( \displaystyle Z = \frac{18000 - 24000}{4000} = -1.5 \)
By symmetry of the normal curve,
\(P(X \lt 18000) = P(Z \lt -1.5)\)
\(= 0.5 - 0.4332 = 0.0668\)
If N is the total number of workers, then 0.0668 N workers earn less than Rs. 18000:
\(0.0668\,N = 334\)
\( \displaystyle N = \frac{334}{0.0668} = 5000 \)
Why the other options are incorrect
Option (1) 10000 — Incorrect. 6.68% of 10000 is 668 workers, not 334.
Option (2) 771 — Incorrect. This is 334 ÷ 0.4332; 0.4332 is the area between Rs. 18000 and the mean, not below Rs. 18000.
Option (4) 2000 — Incorrect. 6.68% of 2000 is about 134 workers, not 334.