Question

A production manager at a wall clock company wants to test their new wall clocks. The designer claims they have a mean life of 16 years with a standard deviation of 4 years. If the claim is true, in a sample of 42 wall clocks, what is the probability that the mean clock life would be greater than 15.1 years? Round your answer to four decimal places.

Answer 1

92.79% probability that the mean clock life would be greater than 15.1 years.

Explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean
`\mu` and standard deviation
`\sigma`, a large sample size can be approximated to a normal distribution with mean
`\mu` and standard deviation
`(\sigma)/(√(n))`.

Normal probability distribution

Problems of normally distributed samples can be 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.

In this problem, we have that:

`\mu = 16, \sigma = 4, n = 42, s = (4)/(√(42)) = 0.6172`

What is the probability that the mean clock life would be greater than 15.1 years?

This probability is 1 subtracted by the pvalue of Z when
`X = 15.1`. So:

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

`Z = (15.1 - 16)/(0.6172)`

`Z = -1.46`

`Z = -1.46` has a pvalue of 0.0721.

So there is a 1-0.0721 = 0.9279 = 92.79% probability that the mean clock life would be greater than 15.1 years.