Question

Manufacturer knows that their items have a normally distributed lifespan, with a mean of 13.9 years, and standard deviation of 3.7 years.

The 3% of items with the shortest lifespan will last less than how many years?
Give your answer to one decimal place.

Answer 1

To find the number of years that the 3% of items with the shortest lifespan will last, we utilize the standard normal distribution and the formula z = (x - mean) / standard deviation. The number of years is approximately 6.29.

Step-by-step explanation:

To find the number of years that the 3% of items with the shortest lifespan will last, we need to find the value corresponding to the lower 3% in a standard normal distribution.

First, we need to calculate the z-score using the formula z = (x - mean) / standard deviation. Substituting the values given in the question, we have z = (x - 13.9) / 3.7.

Next, we find the z-value corresponding to the lower 3% using a z-table or a calculator. The z-value is approximately -1.88.

Finally, we solve for x using the formula x = z * standard deviation + mean. Substituting the values, we have x = -1.88 * 3.7 + 13.9 = 6.29 years.