Finding the Lifespan Percentile for Items with a Normally Distributed Lifespan

A manufacturer knows that their items have a normally distributed lifespan with a mean of 7 years and a standard deviation of 0.9 years. This article discusses how to determine the lifespan that corresponds to the shortest 10% of items using statistical methods, including the z-score formula and the properties of the normal distribution.

Understanding Normal Distribution and Lifespan

In statistics, a normal distribution, or Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that most of the observations cluster around the central value. For our manufacturer, the lifespan of items follows a normal distribution with a mean (μ) of 7 years and a standard deviation (σ) of 0.9 years. This means that most items will last around 7 years, with a certain variability represented by the standard deviation.

Determining the 10th Percentile for Lifespan

The question at hand is to find the lifespan that corresponds to the shortest 10% of items in the distribution. To do this, we can use the z-score formula and the properties of the normal distribution. The z-score is a measure of how many standard deviations an element is from the mean.

Using the Z-Score for the 10th Percentile

The z-score that corresponds to the 10th percentile can be found using a z-table or a standard normal distribution calculator. The z-score for the 10th percentile is approximately -1.28.

Applying the Z-Score Formula

The z-score formula is:

z frac{X - mu}{sigma}

Where:

z is the value we are trying to find mu is the mean (7 years) sigma is the standard deviation (0.9 years)

Rearranging the formula to solve for (X):

X mu z cdot sigma

Plugging in the Values

Now, let's calculate the lifespan:

X 7 (-1.28) cdot 0.9

X 7 - 1.152

X approx 5.848

Therefore, the 10% of items with the shortest lifespan will last less than approximately 5.85 years.

Conclusion

By understanding the properties of the normal distribution and using the z-score formula, manufacturers can predict and manage the expected lifespan of their products. This information is crucial for quality control, product design, and setting reasonable warranties.

For more detailed information on normal distributions and z-scores, refer to statistical textbooks or online resources. This article provides a basic overview and practical application of these concepts in the context of item lifespans.

Keywords: normal distribution, lifespan percentile, z-score calculation