Probability Analysis of Light Bulb Lifespan Using Normal Distribution

Laboratory tests indicate that the lifespan of light bulbs follows a normal distribution with a mean of 750 hours and a standard deviation of 75 hours. In this article, we will explore how to use this information to calculate the probability that a randomly selected light bulb will last between 600 and 750 hours.

Understanding the Parameters of the Normal Distribution

The normal distribution is a continuous probability distribution that is defined by two parameters: the mean (μ) and the standard deviation (σ).

Mean (μ): 750 hours Standard Deviation (σ): 75 hours

Converting Raw Scores to Z-Scores

To find the probability that a randomly selected light bulb will last between 600 and 750 hours, we first need to convert these raw scores to z-scores. The formula for the z-score is:

[ z frac{X - mu}{sigma} ]

For X 600:

[ z_{600} frac{600 - 750}{75} frac{-150}{75} -2 ]

For X 750:

[ z_{750} frac{750 - 750}{75} frac{0}{75} 0 ]

Using the Standard Normal Distribution Table

The standard normal distribution table provides the probabilities corresponding to different z-scores. We can use this table to find the probability values.

Probability corresponding to z -2: approximately 0.0228 Probability corresponding to z 0: 0.5000

To calculate the probability that a light bulb lasts between 600 and 750 hours, we use the following formula:

[ P(600 le X le 750) P(Z le 0) - P(Z le -2) 0.5000 - 0.0228 0.4772 ]

Thus, the probability that a randomly selected light bulb will last between 600 and 750 hours is approximately 0.4772 or 47.72%.

Discussion on Additional Examples

Let's explore some additional examples using the same principles to further understand the application of the normal distribution in probability calculations.

Example 1: Probability Between 1003 and 1020 Hours

Raw Score (X): 1020 Z-Score: ( frac{1020 - 750}{75} frac{270}{75} 3.6 ) Probability corresponding to z 3.6: approximately 0.9998 Probability of being between 1003 and 1020 hours: 1 - 0.9998 0.0002

In this case, the probability of a light bulb lasting between 1003 and 1020 hours is 0.0002 or 0.02%.

Example 2: Probability Between 1088 and 1020 Hours

Raw Score (X): 1088 Z-Score: ( frac{1088 - 750}{75} frac{338}{75} 4.5067 ) Probability corresponding to z 4.5067: approximately 1.0000 Probability of being between 1088 and 1020 hours: 1 - 0.9998 0.0002

Here, the probability of a light bulb lasting between 1088 and 1020 hours is also 0.0002 or 0.02%.

Combining the Results

To combine the results of the two examples, we calculate:

[ 0.0002 0.0002 0.0004 ]

Thus, the combined probability of a light bulb lasting between 1003 and 1088 hours is 0.0004 or 0.04%.

Conclusion

Understanding the normal distribution is crucial for analyzing data in various fields, including reliability engineering. By leveraging the principles of mean and standard deviation, we can calculate probabilities for specific ranges of values. This article demonstrates how to use the normal distribution to estimate the likelihood of a light bulb lasting within a certain time frame.

Keywords: Normal Distribution, Standard Deviation, Probability Calculation