Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 11 years, and standard deviation of 2.7 years.

The 10% of items with the shortest lifespan will last less than how many years?

Give your answer to one decimal place.

Answer 1

It is given that a manufacturer knows that their items have a normally distributed lifespan, with a mean of 11 years, and standard deviation of 2.7 years. Thus, Mean, = 11 and Standard Deviation, = 2.7.

Now, first of all, we need to use the standard normal table to find out which value of z will have 10% of the area (i.e., probability) to the left of it. We can see that z value is approximately equal to -1.282 .[As, in the question, it is said that the shortest lifespan will last less than how many years].

Thus, z = -1.282, = 2.7, = 11.

We know, z = (x - )/

Thus, -1.282 = (x - 11)/2.7

or, (-1.282)*2.7 = x - 11

or, -3.4614 = x - 11

or, x - 11 = - 3.4614

or, x = 11 - 3.4614 = 7.5386

Thus, x = 7.5386 = 7.5(rounded up to one decimal place).

Thus, the 10% of items with the shortest lifespan will last less than 7.5 years.