Respuesta :
Answer:
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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:
[tex]\mu = 16, \sigma = 4, n = 42, s = \frac{4}{\sqrt{42}} = 0.6172[/tex]
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 [tex]X = 15.1[/tex]. So:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{15.1 - 16}{0.6172}[/tex]
[tex]Z = -1.46[/tex]
[tex]Z = -1.46[/tex] 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.
