Model Answer
0 min readIntroduction
In operations management, understanding the lifespan of components and implementing effective replacement strategies is crucial for minimizing downtime and costs. The reliability of products is often modeled using probability distributions, with the normal distribution being a common choice due to its mathematical properties and applicability to many real-world phenomena. This question presents a scenario involving the lifespan of paraffin bulbs, requiring the application of normal distribution principles to determine the number of bulbs likely to fail prematurely and the optimal replacement interval to maintain a desired level of operational reliability.
Part 1: Number of Bulbs Expiring in Less Than 90 Days
We are given that the average life of a bulb is 120 days (μ = 120) with a standard deviation of 20 days (σ = 20). We want to find the number of bulbs that will expire in less than 90 days. This requires calculating the probability of a bulb failing before 90 days and then multiplying that probability by the total number of bulbs (1000).
Step 1: Calculate the Z-score
The Z-score represents the number of standard deviations a particular value is away from the mean. The formula for the Z-score is:
Z = (X - μ) / σ
Where:
- X = The value we are interested in (90 days)
- μ = The mean (120 days)
- σ = The standard deviation (20 days)
Z = (90 - 120) / 20 = -1.5
Step 2: Find the Probability from the Z-table
Using a standard normal distribution table (or a statistical calculator), we find the probability associated with a Z-score of -1.5. This probability represents the proportion of bulbs that will expire in less than 90 days. P(Z < -1.5) ≈ 0.0668
Step 3: Calculate the Number of Bulbs
Number of bulbs expiring in less than 90 days = Total number of bulbs * Probability
Number of bulbs = 1000 * 0.0668 = 66.8
Since we can't have a fraction of a bulb, we round to the nearest whole number. Approximately 67 bulbs will expire in less than 90 days.
Part 2: Replacement Interval for 10% Failure Rate
We want to find the interval between replacements such that no more than 10% of the bulbs expire before replacement. This means we want to find a time 't' such that P(life < t) = 0.10.
Step 1: Find the Z-score for 10% Probability
We need to find the Z-score corresponding to a cumulative probability of 0.10. Using a standard normal distribution table (or a statistical calculator), we find that the Z-score is approximately -1.28.
Step 2: Calculate the Time 't'
We can use the Z-score formula to solve for 't':
Z = (t - μ) / σ
-1.28 = (t - 120) / 20
t - 120 = -1.28 * 20
t - 120 = -25.6
t = 120 - 25.6 = 94.4 days
Therefore, the replacement interval should be approximately 94.4 days to ensure that not more than 10% of the bulbs expire before replacement.
Conclusion
In conclusion, approximately 67 bulbs are expected to expire within the first 90 days of operation. To maintain a failure rate of no more than 10%, the bulbs should be replaced every 94.4 days. These calculations demonstrate the practical application of normal distribution principles in inventory management and reliability engineering. Regular monitoring of bulb lifespan and adjustments to the replacement interval may be necessary to account for variations in manufacturing quality or operating conditions.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.