Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 10.1 years and a standard deviation of 2.4 years. If you randomly select 16 items, what is the probability that their mean lifespan is less than 9 years?

Answer 1

The probability that the mean lifespan of a sample of 16 items is less than 9 years, with the population having a mean lifespan of 10.1 years and a standard deviation of 2.4 years, is approximately 0.034.

Step-by-step explanation:

The question involves finding the probability that the mean lifespan of a sample of items is less than a given value when the lifespan is normally distributed. Given that the mean lifespan is 10.1 years with a standard deviation of 2.4 years, and 16 items are randomly selected, we'll use the standard normal distribution to find this probability.

First, we need to compute the standard error of the mean (SEM), which is the standard deviation divided by the square root of the sample size (n).SEM = 2.4 / sqrt(16) = 2.4 / 4 = 0.6 years.

Next, we calculate the z-score for a mean lifespan of 9 years.Z = (X - μ) / SEMZ = (9 - 10.1) / 0.6Z = (1.1 / 0.6)Z = -1.83.

Finally, we look up the z-score on a standard normal distribution table or use a calculator to find the probability. The probability associated with a z-score of -1.83 is approximately 0.034. This is the probability that the mean lifespan of the 16 items is less than 9 years.