A manufacturer knows that their items have a normally distributed lifespan, with a mean of 14.7 years, and standard deviation of 2.8 years. If you randomly purchase one item, what is the probability it will last longer than 20 years? Question Help: Video Submit Question - Previous

Step 1 ：The problem is asking for the probability that a randomly selected item will last longer than 20 years. This is a problem of normal distribution.

Step 2 ：We can solve this problem by calculating the z-score of 20 years and then finding the area to the right of this z-score in the standard normal distribution.

Step 3 ：The z-score is calculated as follows: \(z = \frac{X - \mu}{\sigma}\) where X is the value we are interested in (20 years), \(\mu\) is the mean (14.7 years), and \(\sigma\) is the standard deviation (2.8 years).

Step 4 ：After calculating the z-score, we can find the probability by looking up the z-score in a standard normal distribution table or using a function that calculates this probability.

Step 5 ：However, this will give us the probability that an item will last less than 20 years. To find the probability that an item will last more than 20 years, we need to subtract this probability from 1.

Step 6 ：Using the given values, we find that the z-score is approximately 1.89.

Step 7 ：Looking up this z-score in a standard normal distribution table, we find that the probability that an item will last less than 20 years is approximately 0.97.

Step 8 ：Subtracting this probability from 1, we find that the probability that an item will last more than 20 years is approximately 0.029.

Step 9 ：Final Answer: The probability that a randomly selected item will last longer than 20 years is approximately \(\boxed{0.029}\).