The Life Spans Of A Computer Manufacturer's Hard Drives Are Normally Distributed, With A Mean Of 3 Years 6 Months And A Standard Deviation Of 9 Months. What Is The Probability Of A Randomly Selected Hard Drive From The Company Lasting Between 2 Years 3

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Introduction

In the world of computer manufacturing, the lifespan of hard drives is a crucial factor that affects the overall performance and reliability of a system. Understanding the probability distribution of hard drive lifespans can help manufacturers make informed decisions about product design, quality control, and customer expectations. In this article, we will explore the normal distribution of hard drive lifespans and calculate the probability of a randomly selected hard drive lasting between 2 years 3 months and 4 years 9 months.

Understanding Normal Distribution

A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In the context of hard drive lifespans, the mean represents the average lifespan of a hard drive, while the standard deviation represents the amount of variation or dispersion from the mean.

Given Information

  • Mean (μ) = 3 years 6 months = 42 months
  • Standard Deviation (σ) = 9 months

Calculating Probability

To calculate the probability of a hard drive lasting between 2 years 3 months and 4 years 9 months, we need to convert these time periods to months and then use the z-score formula to find the corresponding probabilities.

Converting Time Periods to Months

  • 2 years 3 months = 27 months
  • 4 years 9 months = 57 months

Calculating z-Scores

The z-score formula is:

z = (X - μ) / σ

where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.

For X = 27 months:

z = (27 - 42) / 9 z = -15 / 9 z = -1.67

For X = 57 months:

z = (57 - 42) / 9 z = 15 / 9 z = 1.67

Using a z-Table or Calculator

Using a z-table or calculator, we can find the probabilities corresponding to the z-scores.

For z = -1.67:

P(Z < -1.67) = 0.0475

For z = 1.67:

P(Z < 1.67) = 0.9525

Finding the Probability of a Hard Drive Lasting Between 2 Years 3 Months and 4 Years 9 Months

To find the probability of a hard drive lasting between 2 years 3 months and 4 years 9 months, we need to subtract the probability of a hard drive lasting less than 2 years 3 months from the probability of a hard drive lasting less than 4 years 9 months.

P(27 < X < 57) = P(Z < 1.67) - P(Z < -1.67) P(27 < X < 57) = 0.9525 - 0.0475 P(27 < X < 57) = 0.905

Therefore, the probability of a randomly selected hard drive from the company lasting between 2 years 3 months and 4 years 9 months is approximately 90.5%.

Conclusion

In conclusion, understanding the probability distribution of hard drive lifespans is crucial for computer manufacturers to make informed decisions about product design, quality control, and customer expectations. By using the normal distribution and z-score formula, we can calculate the probability of a hard drive lasting between specific time periods. In this article, we calculated the probability of a hard drive lasting between 2 years 3 months and 4 years 9 months and found that it is approximately 90.5%.

References

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

Further Reading

  • Understanding Normal Distribution: A Comprehensive Guide
  • Calculating z-Scores: A Step-by-Step Guide
  • Probability Distributions: A Brief Overview
    The Life Spans of Computer Manufacturer's Hard Drives: Understanding Probability Distributions - Q&A ===========================================================

Introduction

In our previous article, we explored the normal distribution of hard drive lifespans and calculated the probability of a randomly selected hard drive lasting between 2 years 3 months and 4 years 9 months. In this article, we will answer some frequently asked questions related to the life spans of computer manufacturer's hard drives and probability distributions.

Q&A

Q: What is the average lifespan of a hard drive?

A: The average lifespan of a hard drive is 3 years 6 months, which is equivalent to 42 months.

Q: What is the standard deviation of hard drive lifespans?

A: The standard deviation of hard drive lifespans is 9 months.

Q: What is the probability of a hard drive lasting less than 2 years 3 months?

A: Using the z-score formula, we can calculate the probability of a hard drive lasting less than 2 years 3 months. The z-score is -1.67, and using a z-table or calculator, we can find that the probability of a hard drive lasting less than 2 years 3 months is approximately 4.75%.

Q: What is the probability of a hard drive lasting more than 4 years 9 months?

A: Using the z-score formula, we can calculate the probability of a hard drive lasting more than 4 years 9 months. The z-score is 1.67, and using a z-table or calculator, we can find that the probability of a hard drive lasting more than 4 years 9 months is approximately 95.25%.

Q: How can I use the normal distribution to make informed decisions about product design and quality control?

A: The normal distribution can be used to make informed decisions about product design and quality control by understanding the probability of a hard drive lasting within a specific time period. For example, if a manufacturer wants to design a hard drive that lasts for at least 3 years, they can use the normal distribution to calculate the probability of a hard drive lasting within that time period.

Q: Can I use the normal distribution to calculate the probability of a hard drive failing within a specific time period?

A: Yes, you can use the normal distribution to calculate the probability of a hard drive failing within a specific time period. However, you will need to use the complement rule, which states that the probability of a hard drive failing within a specific time period is equal to 1 minus the probability of a hard drive lasting within that time period.

Q: How can I use the z-score formula to calculate the probability of a hard drive lasting within a specific time period?

A: The z-score formula is:

z = (X - μ) / σ

where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation. Once you have calculated the z-score, you can use a z-table or calculator to find the corresponding probability.

Conclusion

In conclusion, understanding the probability distribution of hard drive lifespans is crucial for computer manufacturers to make informed decisions about product design, quality control, and customer expectations. By using the normal distribution and z-score formula, we can calculate the probability of a hard drive lasting within a specific time period. We hope that this Q&A article has provided you with a better understanding of the life spans of computer manufacturer's hard drives and probability distributions.

References

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

Further Reading

  • Understanding Normal Distribution: A Comprehensive Guide
  • Calculating z-Scores: A Step-by-Step Guide
  • Probability Distributions: A Brief Overview