A manufacturer knows that their items have a normally distributed lifespan, with a mean of 11.9 years, and standard deviation of 2.2 years.

If you randomly purchase one item, what is the probability it will last longer than 18 years?

To find the probability that an item will last longer than 18 years, we need to calculate the area under the curve of the normal distribution to the right of 18.

First, we need to standardize the value of 18 using the z-score formula:

z = (x - mean) / standard deviation

where x is the value we want to standardize, mean is the mean of the distribution, and standard deviation is the standard deviation of the distribution.

In this case:
x = 18
mean = 11.9 years
standard deviation = 2.2 years

Calculating the z-score:
z = (18 - 11.9) / 2.2
z = 2.7727

Next, we need to find the area under the curve to the right of the z-score using a standard normal distribution table or a calculator.

Using a standard normal distribution table, the area to the right of z = 2.7727 is approximately 0.002955.

So, the probability that a randomly purchased item will last longer than 18 years is approximately 0.002955, or 0.2955%.