Understanding the Lifespan of Light Bulbs Through Normal Distribution

An electrical firm manufactures light bulbs whose lifespans, before burn-out, follow a normally distributed pattern with a mean lifespan of 800 hours and a standard deviation of 40 hours. This article aims to illustrate how to calculate the probability that a light bulb will last between 778 and 834 hours, using the principles of normal distribution in statistics.

Introduction

Lifespan of light bulbs, particularly those produced by an electrical firm, is often of interest to consumers and manufacturers alike. Assuming a normal distribution for the lifespan of these light bulbs, the mean (μ) is 800 hours, and the standard deviation (σ) is 40 hours. This distribution allows us to apply statistical methods to determine the likelihood of various outcomes in the lifespan of these light bulbs.

Steps to Calculate the Probability

The process of finding the probability that a light bulb will last between 778 and 834 hours involves calculating the given Z-scores and then using these scores to find the corresponding probability using the normal distribution table or statistical software.

Calculating Z-Scores

The first step is to calculate the Z-scores for 778 hours and 834 hours. The formula for calculating a Z-score is:

Z (X - μ) / σ

For 778 Hours:

Z (778 - 800) / 40

Z -22 / 40 -0.55

For 834 Hours:

Z (834 - 800) / 40

Z 34 / 40 0.85

Using the Normal Distribution Table

With the Z-scores calculated, we can now find the corresponding probabilities for each Z-value. The probability between two Z-scores, -0.55 and 0.85, can be found by looking up the values in a standard normal distribution table or using a statistical software like R, which has a library called 'tigerstats'.

Using R and 'tigerstats'

The R code to calculate the probability using the 'tigerstats' package is as follows:

library(tigerstats)pnormGCc(778, 834, region  "between")

The output from the R code is:

[1] 0.5111778

This indicates that the probability that a light bulb lasts between 778 and 834 hours is approximately 0.5112 or 51.12%.

Conclusion

The use of normal distribution allows us to understand the probabilistic nature of the lifespan of light bulbs produced by the electrical firm. In this case, it is found that there is a 51.12% probability that a light bulb will last between 778 and 834 hours. This information can be valuable for various stakeholders, including manufacturers, retailers, and consumers, in understanding the expected lifespan of these light bulbs.

References

To cite package ‘tigerstats’ in publications use: Rebekah Robinson and Homer White (2020). tigerstats: R Functions for Elementary Statistics. R package version 0.3.2.