Respuesta :
Answer:
She should offer a guarantee of 6.24 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 problem, we have that:
[tex]\mu = 10, \sigma = 2[/tex]
If she is willing to replace 3% of the motors that fail, how long a guarantee (in years) should she offer?
This is the value of X when Z has a pvalue of 0.03. So it is X when Z = -1.88. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.88 = \frac{X - 10}{2}[/tex]
[tex]X - 10 = -1.88*2[/tex]
[tex]X = 6.24[/tex]
She should offer a guarantee of 6.24 years.
