Questions: A manufacturer knows that their items have a normally distributed lifespan, with a mean of 12.2 years, and standard deviation of 1.6 years. The 6% of items with the shortest lifespan will last less than how many years? Give your answer to one decimal place.

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 12.2 years, and standard deviation of 1.6 years.

The 6% of items with the shortest lifespan will last less than how many years? Give your answer to one decimal place.

Transcript text: A manufacturer knows that their items have a normally distributed lifespan, with a mean of 12.2 years, and standard deviation of 1.6 years. The $6 \%$ of items with the shortest lifespan will last less than how many years? Give your answer to one decimal place. Question Help: Video Post to forum Submit Question

Solution

Solution Steps

Step 1: Define the Normal Distribution Parameters

The lifespan of the items is normally distributed with the following parameters:

Mean ($ \mu $): $ 12.2 $ years
Standard Deviation ($ \sigma $): $ 1.6 $ years

Step 2: Identify the Desired Percentile

We are interested in finding the lifespan below which $ 6\% $ of the items fall. This corresponds to the $ 0.06 $ quantile of the normal distribution.

Step 3: Calculate the Inverse CDF

To find the lifespan corresponding to the $ 6\% $ percentile, we use the inverse cumulative distribution function (CDF) for a normal distribution: \[ x = \mu + z \cdot \sigma \] where $ z $ is the z-score corresponding to the $ 0.06 $ quantile.

Step 4: Find the Z-Score

Using the properties of the normal distribution, we find the z-score $ z $ such that: \[ P(X \leq x) = 0.06 \]

Step 5: Calculate the Lifespan

Substituting the values into the equation, we find: \[ x = 12.2 + z \cdot 1.6 \] After calculating, we find that $ x $ is approximately $ 9.7 $ years.

Step 6: Conclusion

Thus, $ 6\% $ of the items with the shortest lifespan will last less than $ 9.7 $ years.