The Normal Distribution of Light Bulb Lifespan

Oct 13, 2025 by miodeya.com 90 views

Introduction

In the world of statistics, the normal distribution is a fundamental concept that helps us understand and analyze various phenomena. In this article, we will delve into the world of light bulbs and explore how their lifespan follows a normal distribution. We will use the $$68%-95%-99.7%$$ rule to find the probability that a given light bulb lasts between 675 and 900 hours.

The Normal Distribution of Light Bulb Lifespan

The lifespan of a light bulb is a random variable that follows a normal distribution with a mean (μ) of 750 hours and a standard deviation (σ) of 75 hours. This means that the lifespan of a light bulb can be represented by a bell-shaped curve, with the majority of the data points clustered around the mean.

The $$68%-95%-99.7%$$ Rule

The $$68%-95%-99.7%$$ rule, also known as the empirical rule, states that in a normal distribution:

About 68% of the data points fall within 1 standard deviation of the mean (μ ± σ)
About 95% of the data points fall within 2 standard deviations of the mean (μ ± 2σ)
About 99.7% of the data points fall within 3 standard deviations of the mean (μ ± 3σ)

Finding the Probability

To find the probability that a given light bulb lasts between 675 and 900 hours, we need to calculate the z-scores for these values.

Calculating the z-scores

The z-score is a measure of how many standard deviations an observation is away from the mean. It can be calculated using the following formula:

z = (X - μ) / σ

where X is the value of the observation, μ is the mean, and σ is the standard deviation.

For X = 675, we have:

z = (675 - 750) / 75 = -75 / 75 = -1

For X = 900, we have:

z = (900 - 750) / 75 = 150 / 75 = 2

Using the z-scores to find the probability

Now that we have the z-scores, we can use a standard normal distribution table (also known as a z-table) to find the probability that a given light bulb lasts between 675 and 900 hours.

Finding the probability using the z-table

Using the z-table, we find that:

P(Z < -1) = 0.1587 (this is the area to the left of z = -1)
P(Z < 2) = 0.9772 (this is the area to the left of z = 2)

Calculating the probability

To find the probability that a given light bulb lasts between 675 and 900 hours, we need to subtract the area to the left of z = -1 from the area to the left of z = 2.

P(675 < X < 900) = P(Z < 2) - P(Z < -1) = 0.9772 - 0.1587 = 0.8185

Conclusion

In this article, we used the $$68%-95%-99.7%$$ rule to find the probability that a given light bulb lasts between 675 and 900 hours. We calculated the z-scores for these values and used a standard normal distribution table to find the probability. The result shows that there is a 81.85% chance that a given light bulb will last between 675 and 900 hours.

Discussion

The normal distribution of light bulb lifespan is a useful tool for manufacturers and consumers alike. By understanding the probability of a light bulb lasting a certain amount of time, manufacturers can design their products to meet the needs of their customers. Consumers, on the other hand, can use this information to make informed decisions about which light bulbs to purchase.

Limitations

While the normal distribution of light bulb lifespan is a useful tool, it is not without its limitations. For example, the lifespan of a light bulb can be affected by various factors such as temperature, humidity, and usage patterns. Therefore, the probability calculated in this article may not be accurate in all cases.

Future Research

Future research could focus on developing more accurate models of light bulb lifespan, taking into account various factors that can affect its duration. Additionally, researchers could explore the use of other statistical distributions, such as the Weibull distribution, to model light bulb lifespan.

References

[1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: The exploration and analysis of data. John Wiley & Sons.

Appendix

The following is a list of formulas and tables used in this article:

z = (X - μ) / σ
P(Z < z) = 0.1587 (z = -1)
P(Z < z) = 0.9772 (z = 2)

Introduction

In our previous article, we explored the normal distribution of light bulb lifespan and used the $$68%-95%-99.7%$$ rule to find the probability that a given light bulb lasts between 675 and 900 hours. In this article, we will answer some frequently asked questions about the normal distribution of light bulb lifespan.

Q: What is the normal distribution of light bulb lifespan?

A: The normal distribution of light bulb lifespan is a statistical distribution that describes the lifespan of a light bulb. It is a bell-shaped curve that shows the probability of a light bulb lasting a certain amount of time.

Q: What is the mean and standard deviation of the normal distribution of light bulb lifespan?

A: The mean of the normal distribution of light bulb lifespan is 750 hours, and the standard deviation is 75 hours.

Q: How is the probability of a light bulb lasting a certain amount of time calculated?

A: The probability of a light bulb lasting a certain amount of time is calculated using the z-score formula: z = (X - μ) / σ, where X is the value of the observation, μ is the mean, and σ is the standard deviation. The z-score is then used to find the probability using a standard normal distribution table.

Q: What is the $$68%-95%-99.7%$$ rule?

A: The $$68%-95%-99.7%$$ rule, also known as the empirical rule, states that in a normal distribution:

About 68% of the data points fall within 1 standard deviation of the mean (μ ± σ)
About 95% of the data points fall within 2 standard deviations of the mean (μ ± 2σ)
About 99.7% of the data points fall within 3 standard deviations of the mean (μ ± 3σ)

Q: How can the normal distribution of light bulb lifespan be used in real-world applications?

A: The normal distribution of light bulb lifespan can be used in various real-world applications, such as:

Designing light bulbs that meet the needs of consumers
Predicting the lifespan of light bulbs in different environments
Developing maintenance schedules for light bulbs
Estimating the cost of replacing light bulbs

Q: What are some limitations of the normal distribution of light bulb lifespan?

A: Some limitations of the normal distribution of light bulb lifespan include:

The lifespan of a light bulb can be affected by various factors such as temperature, humidity, and usage patterns
The normal distribution may not accurately model the lifespan of light bulbs in certain environments
The z-score formula may not be accurate for all values of X

Q: What are some future research directions for the normal distribution of light bulb lifespan?

A: Some future research directions for the normal distribution of light bulb lifespan include:

Developing more accurate models of light bulb lifespan that take into account various factors that can affect its duration
Exploring the use of other statistical distributions, such as the Weibull distribution, to model light bulb lifespan
Investigating the impact of environmental factors on the lifespan of light bulbs

Conclusion

In this article, we answered some frequently asked questions about the normal distribution of light bulb lifespan. We hope that this information will be helpful to those who are interested in understanding the statistical distribution of light bulb lifespan.

References

[1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: The exploration and analysis of data. John Wiley & Sons.

Appendix

The following is a list of formulas and tables used in this article:

z = (X - μ) / σ
P(Z < z) = 0.1587 (z = -1)
P(Z < z) = 0.9772 (z = 2)

Note: The z-table used in this article is a standard normal distribution table, which can be found in most statistics textbooks or online resources.