Amanufacturer knows that their items have a normally distributed lifespan, with a mean of 13.2 years, and standard deviation of 3.1 years. if you randomly purchase 12 items, what is the probability that their mean life will be longer than 12 years?

There is a 91.15% probability that their mean life will be longer than 12 years.

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 13.2 years, and standard deviation of 3.1 years. This means that .

If you randomly purchase 12 items, what is the probability that their mean life will be longer than 12 years?

There are 12 items, so .

This is 1 subtracted by the pvalue of Z when .

By the Central Limit Theorem, we use the standard deviation of the sample in the Z score formula. That is:

has a pvalue of 0.0885

This means that there is a 1-0.0885 = 0.9115 = 91.15% probability that their mean life will be longer than 12 years.

n

step-by-step explanation: