The Times Taken To Assemble A Clock At A Factory Are Approximately Normally Distributed With A Mean $\mu = 3 \, \text{hr}$ And A Standard Deviation $\sigma = 0.5 \, \text{hr}$. What Percentage Of The Times Are Between 2 Hr And 4 Hr?A.

Introduction

In a factory, the time taken to assemble a clock is a critical parameter that affects the overall productivity and efficiency of the manufacturing process. Understanding the distribution of this time is essential for making informed decisions and optimizing the production process. In this article, we will analyze the time taken to assemble a clock at a factory, which is approximately normally distributed with a mean $\mu = 3 \, \text{hr}$ and a standard deviation $\sigma = 0.5 \, \text{hr}$. We will calculate the percentage of times that are between 2 hr and 4 hr.

Understanding the Normal Distribution

The normal distribution, also known as the 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 this case, the mean time taken to assemble a clock is 3 hr, and the standard deviation is 0.5 hr. This means that most of the times taken to assemble a clock will be close to 3 hr, with fewer times taken at the extremes.

Calculating the Percentage of Times Between 2 hr and 4 hr

To calculate the percentage of times between 2 hr and 4 hr, we need to use the z-score formula:

$$z = \frac{X - \mu}{\sigma}$$

where $X$ is the time taken to assemble a clock, $\mu$ is the mean time, and $\sigma$ is the standard deviation.

For $X = 2 \, \text{hr}$, we have:

$$z = \frac{2 - 3}{0.5} = -2$$

For $X = 4 \, \text{hr}$, we have:

$$z = \frac{4 - 3}{0.5} = 2$$

Using a Standard Normal Distribution Table

We can use a standard normal distribution table to find the probabilities corresponding to the z-scores. The table shows the probability that a standard normal variable $Z$ takes on a value less than or equal to $z$. We can use this table to find the probabilities corresponding to the z-scores we calculated earlier.

For $z = -2$, the probability is approximately 0.0228. This means that about 2.28% of the times taken to assemble a clock are less than 2 hr.

For $z = 2$, the probability is approximately 0.9772. This means that about 97.72% of the times taken to assemble a clock are less than 4 hr.

Calculating the Percentage of Times Between 2 hr and 4 hr

To calculate the percentage of times between 2 hr and 4 hr, we need to subtract the probability of times less than 2 hr from the probability of times less than 4 hr:

$$\text{Percentage} = 97.72\% - 2.28\% = 95.44\%$$

Therefore, approximately 95.44% of the times taken to assemble a clock are between 2 hr and 4 hr.

Conclusion

In this article, we analyzed the time taken to assemble a clock at a factory, which is approximately normally distributed with a mean $\mu = 3 \, \text{hr}$ and a standard deviation $\sigma = 0.5 \, \text{hr}$. We calculated the percentage of times that are between 2 hr and 4 hr, which is approximately 95.44%. This information can be used to optimize the production process and improve the overall efficiency of the factory.

References

• [1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
• [2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.

Additional Resources

• [1] Khan Academy. (n.d.). Normal distribution. Retrieved from https://www.khanacademy.org/math/statistics-probability/normal_distribution
• [2] Stat Trek. (n.d.). Normal distribution. Retrieved from https://stattrek.com/distributions/normal.aspx
  

Introduction

In our previous article, we analyzed the time taken to assemble a clock at a factory, which is approximately normally distributed with a mean $\mu = 3 \, \text{hr}$ and a standard deviation $\sigma = 0.5 \, \text{hr}$. We calculated the percentage of times that are between 2 hr and 4 hr, which is approximately 95.44%. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the normal distribution?

A: The normal distribution, also known as the 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.

Q: What is the mean time taken to assemble a clock?

A: The mean time taken to assemble a clock is 3 hr.

Q: What is the standard deviation of the time taken to assemble a clock?

A: The standard deviation of the time taken to assemble a clock is 0.5 hr.

Q: How do I calculate the percentage of times between 2 hr and 4 hr?

A: To calculate the percentage of times between 2 hr and 4 hr, you need to use the z-score formula:

$$z = \frac{X - \mu}{\sigma}$$

where $X$ is the time taken to assemble a clock, $\mu$ is the mean time, and $\sigma$ is the standard deviation.

For $X = 2 \, \text{hr}$, we have:

$$z = \frac{2 - 3}{0.5} = -2$$

For $X = 4 \, \text{hr}$, we have:

$$z = \frac{4 - 3}{0.5} = 2$$

You can then use a standard normal distribution table to find the probabilities corresponding to the z-scores.

Q: What is the probability of times less than 2 hr?

A: The probability of times less than 2 hr is approximately 0.0228.

Q: What is the probability of times less than 4 hr?

A: The probability of times less than 4 hr is approximately 0.9772.

Q: How do I calculate the percentage of times between 2 hr and 4 hr?

A: To calculate the percentage of times between 2 hr and 4 hr, you need to subtract the probability of times less than 2 hr from the probability of times less than 4 hr:

$$\text{Percentage} = 97.72\% - 2.28\% = 95.44\%$$

Q: What is the significance of the normal distribution in this context?

A: The normal distribution is significant in this context because it allows us to understand the distribution of the time taken to assemble a clock. This information can be used to optimize the production process and improve the overall efficiency of the factory.

Q: How can I apply this knowledge in real-world scenarios?

A: You can apply this knowledge in real-world scenarios by using statistical analysis to understand the distribution of times taken to complete tasks. This can help you optimize processes, improve efficiency, and make informed decisions.

Conclusion

In this article, we answered some frequently asked questions related to the time taken to assemble a clock at a factory. We provided explanations and examples to help clarify the concepts and calculations involved. We hope this article has been helpful in providing a better understanding of the normal distribution and its applications.

References

• [1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
• [2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.

Additional Resources

• [1] Khan Academy. (n.d.). Normal distribution. Retrieved from https://www.khanacademy.org/math/statistics-probability/normal_distribution
• [2] Stat Trek. (n.d.). Normal distribution. Retrieved from https://stattrek.com/distributions/normal.aspx