Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 5.5 years, and standard deviation of 1.2 years. If 24 items are picked at random, 8% of the time their mean life will be less than how many years?

Answer 1

If 24 items are picked at random, 8% of the time their mean life will be less than 5.156 years.

Explanation:

To solve this question, we have to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

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

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

`Z = (X - \mu)/(\sigma)`

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.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 5.5, \sigma = 1.2, n = 24, s = (1.2)/(√(24)) = 0.2449`

If 24 items are picked at random, 8% of the time their mean life will be less than how many years?

This is the value of X when Z has a pvalue of 0.08. So it is X when Z = -1.405.

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`-1.405 = (X - 5.5)/(0.2449)`

`X - 5.5 = -1.405*0.2449`

`X = 5.156`

If 24 items are picked at random, 8% of the time their mean life will be less than 5.156 years.