Question 13
6 pts
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 13.1 years, and standard deviation of 4.1 years.

The $9 \%$ of items with the shortest lifespan will last less than how many years?

Round your answer to one decimal place.
Submit and End

Step 1 ：Given that the lifespan of the items is normally distributed with a mean of 13.1 years and a standard deviation of 4.1 years.

Step 2 ：We are asked to find the lifespan below which the shortest 9% of items will last.

Step 3 ：This corresponds to finding the lifespan at the 9th percentile of the distribution.

Step 4 ：We can find this by first finding the z-score that corresponds to the 9th percentile, which is approximately -1.34.

Step 5 ：We then convert this z-score to a lifespan in years using the formula \(lifespan = mean + z\_score \times std\_dev\).

Step 6 ：Substituting the given values, we get \(lifespan = 13.1 + (-1.34) \times 4.1\).

Step 7 ：Calculating the above expression, we get a lifespan of approximately 7.6 years.

Step 8 ：Rounding this to one decimal place, we get a lifespan of 7.6 years.

Step 9 ：Final Answer: The 9% of items with the shortest lifespan will last less than \(\boxed{7.6}\) years.