Management at Gordon Electronics is considering adopting a bonus system to increase production. One suggestion is to pay a bonus on the highest 5% of production based on past experience. Past records indicate weekly production follows the normal distribution. The mean of this distribution is 4,000 units per week and the standard deviation is 60 units per week. If the bonus is paid on the upper 5% of production, the bonus will be paid on how many units or more?

The cumulative distribution of a random variable X that follows a normal distribution is given by the area undear the "bell curve" and the values are given by the corresponding table for the standard normal distribution.

The standardized value of the variable X is called Z and is calculated with the formula:

[tex]Z=\dfrac{X-\mu}{\sigma}[/tex]

Where:

[tex]\mu=mean=4,000[/tex]

[tex]\sigma=standard\text{ }deviation=60[/tex]

You read the Z-value for which the probability is greater than or equal to 5% in the table for the values of the area to the right of Z. Using probability = area under the curve ≥ 5%, the Z-value is 1.645 (interpolating between p = 0.0495, Z = 1.64 and p = 0.0505, Z = 1.65).

Substituting in the formula for Z:

1.645 = (X - 4,000) / 60

X= 60 × 1.645 + 4,000 = 4,098.7 ≈ 4,099

Hence, the bonus will be paid on 4,099 units or more.