Answer:
They will last less than 1.257 years.
Step-by-step explanation:
Problems of normally distributed samples are 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 question, we have that:
[tex]\mu = 2.1, \sigma = 0.6[/tex]
The 8% of items with the shortest lifespan will last less than how many years?
Less than the 8th percentile, which is X when Z has a pvalue of 0.08. So X when Z = -1.405.
So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.405 = \frac{X - 2.1}{0.6}[/tex]
[tex]X - 2.1 = -1.405*0.6[/tex]
[tex]X = 1.257[/tex]
They will last less than 1.257 years.
