Question

Optimal-Eats blender has a mean time before failure of 40 months with a standard deviation of 6 months, and the failure times are normally distributed. What should be the warranty period, in months, so that the manufacturer will not have more than 10% of the blenders returned

Answer 1

Answer:

The warranty period should be of 32 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 $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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.

Mean time before failure of 40 months with a standard deviation of 6 months.

This means that $\mu = 40, \sigma = 6$

What should be the warranty period, in months, so that the manufacturer will not have more than 10% of the blenders returned?

The warranty period should be the 10th percentile, which is X when Z has a p-value of 0.1, so X when Z = -1.28.

$Z = \frac{X - \mu}{\sigma}$

$-1.28 = \frac{X - 40}{6}$

$X - 40 = -1.28*6$

$X = 32$

The warranty period should be of 32 months.