Answer:
They should warranty the product for 7 years if they want no more than 6.7% of the waffle irons to fail within that time.
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.
The average waffle iron lasts for 12 years and one standard deviation is 8 months.
Measuring the time in months, we have that
and ![\sigma = 8](https://tex.z-dn.net/?f=%5Csigma%20%3D%208)
How long should they warranty the product for if they want no more than 6.7% of the waffle irons to fail within that time?
This is X when Z has a p-value of 0.067, so X when Z = -1.5. Then
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-1.5 = \frac{X - 96}{8}](https://tex.z-dn.net/?f=-1.5%20%3D%20%5Cfrac%7BX%20-%2096%7D%7B8%7D)
![X - 96 = -1.5*8](https://tex.z-dn.net/?f=X%20-%2096%20%3D%20-1.5%2A8)
![X = 84](https://tex.z-dn.net/?f=X%20%3D%2084)
84 months = 7 years.
They should warranty the product for 7 years if they want no more than 6.7% of the waffle irons to fail within that time.