Mathematics: 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.

Aproduction manager at a wall clock company wants

Ответ:

wafflewarriormg

13.01.2022

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 $\mu$ and standard deviation $\sigma$ , a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$ .

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:

$Z = \frac{X - \mu}{\sigma}$

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:

$\mu = 16, \sigma = 4, n = 42, s = \frac{4}{\sqrt{42}} = 0.6172$