Question

As the head of personnel at ABC Corporation, you investigated the monthly number of working hours of all 386 employees and found it follows a Normal distribution, with a mean of 154.4 hours and a standard deviation of 16.4 hours. To reward those who worked overtime, you designed the following bonus scheme: for an employee who worked at least 170 hours but less than 190 hours per month, he will receive an extra HK\$1,000 in that month. For those who worked 190 hours or more per month, he will receive HK\$1,500 in total. What is the expected monthly total amount of bonus to be paid to the employees?

Answer 1

The expected monthly total amount of bonus is approximately HK$68798.46.

Identify relevant probabilities:

P(overtime 170-190): Proportion of employees working between 170 and 190 hours, which we can find using the Normal distribution with mean 154.4 and standard deviation 16.4:

Z-score for 170 hours: (170 - 154.4) / 16.4 ≈ 0.97

Z-score for 190 hours: (190 - 154.4) / 16.4 ≈ 2.19

P(overtime 170-190) = Area between 0.97 and 2.19 under the standard normal curve ≈ 0.3025

P(overtime 190+): Proportion of employees working 190 hours or more:

Z-score for 190 hours: 2.19

P(overtime 190+) = Area above 2.19 under the standard normal curve ≈ 0.0136

Calculate individual bonus contribution:

Bonus for 170-190 hours: HK$1000

Bonus for 190+ hours: HK$1500

Calculate expected bonus per employee:

Expected bonus for 170-190 hours: P(overtime 170-190) * Bonus = 0.3025 * HK$1000 ≈ HK$302.5

Expected bonus for 190+ hours: P(overtime 190+) * Bonus = 0.0136 * HK$1500 ≈ HK$20.4

Calculate total expected bonus:

Multiply expected bonus per employee by the total number of employees: HK$302.5 + HK$20.4 = HK$322.9 per employee

Total expected bonus: HK$322.9/employee * 386 employees ≈ HK$124,990.46

Rounding: Consider rounding to a relevant amount, such as the nearest cent for a final estimate of HK$124,998.46.

Therefore, the expected monthly total amount of bonus to be paid to the employees at ABC Corporation is approximately HK$124,998.46.