Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 4 years. and standard deviation of 1.3 years. The 2% 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 2% of items with the shortest lifespan will last, use the z-score formula and solve for x.

Step-by-step explanation:

To find the number of years that the 2% of items with the shortest lifespan will last, we need to find the z-score associated with the 2nd percentile of the normal distribution. The z-score can be calculated using the formula:

z = (x - mean) / standard deviation

Substituting the given values, we have:

z = (x - 4) / 1.3

Since we’re looking for the shortest lifespan, the z-score will be negative. We can use a z-score table or a calculator to find the z-score corresponding to the 2nd percentile, which is approximately -2.05.

Now, we can solve for x:

-2.05 = (x - 4) / 1.3

Dividing by 1.3 and rearranging the equation, we get:

x = -2.05 * 1.3 + 4

Simplifying, we find that x is approximately 1.37.

Therefore, the 2% of items with the shortest lifespan will last less than 1.4 years.