Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 10.8 years, and standard deviation of 3.4 years.

If you randomly purchase 20 items, what is the probability that their mean life will be longer than 11 years?

Answer 1

To calculate the probability that the mean life of 20 items exceeds 11 years, determine the standard error, compute the z-score for a mean of 11 years, and consult the standard normal distribution for the corresponding probability.

Step-by-step explanation:

To find the probability that the mean life of 20 randomly purchased items will be longer than 11 years, given that the lifespans are normally distributed with a mean of 10.8 years and a standard deviation of 3.4 years, we can use the Central Limit Theorem. The standard error (SE) of the sample mean is the population standard deviation divided by the square root of the sample size, or SE = 3.4 / √20. We then calculate the z-score for a sample mean of 11 years, which is (11 - 10.8) / SE. Finally, we find the probability corresponding to this z-score from the standard normal distribution table or using a calculator with normal distribution functions.

The step-by-step process is:

1. Calculate the standard error (SE): SE = 3.4 / √20.
2. Compute the z-score for a mean life of 11 years: z = (11 - 10.8) / SE.
3. Look up the probability associated with the z-score in the standard normal distribution table, or use a calculator.
4. The result will give the probability that the sample mean life of 20 items will be greater than 11 years.