Aproduction 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.

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

Normal probability distribution

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:

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 . So:

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.