Answer:
The warranty period should be of 35 months.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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 p-value, we get the probability that the value of the measure is greater than X.
Cook-Easy blender has a mean time before failure of 41 months with a standard deviation of 4 months
This means that ![\mu = 41, \sigma = 4](https://tex.z-dn.net/?f=%5Cmu%20%3D%2041%2C%20%5Csigma%20%3D%204)
What should be the warranty period, in months, so that the manufacturer will not have more than 8% of the blenders returned?
The warranty period should be the 8th percentile, which is X when Z has a p-value of 0.08, so X when Z = -1.405.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-1.405 = \frac{X - 41}{4}](https://tex.z-dn.net/?f=-1.405%20%3D%20%5Cfrac%7BX%20-%2041%7D%7B4%7D)
![X - 41 = -1.405*4](https://tex.z-dn.net/?f=X%20-%2041%20%3D%20-1.405%2A4)
![X = 35.4](https://tex.z-dn.net/?f=X%20%3D%2035.4)
Rounding to the nearest integer, 35.
The warranty period should be of 35 months.