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Zinaida [17]
3 years ago
5

The average life of a certain type of small motor is 10 years with a standard deviation of 2 years. The manufacturer replaces fr

ee all motors that fail while under guarantee. If she is willing to replace only 3% of the motors that fail, how long a guarantee should be offered? Assume that the lifetime of a motor follows a normal distribution.
Mathematics
1 answer:
taurus [48]3 years ago
7 0

Answer:

A guarantee of 6.24 years should be offered.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

\mu = 10, \sigma = 2

If she is willing to replace only 3% of the motors that fail, how long a guarantee should be offered?

The 3rd percentile, that is, the value of X when Z has a pvalue of 0.03. So X when Z = -1.88.

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

-1.88 = \frac{X - 10}{2}

X - 10 = -1.88*2

X = 6.24

A guarantee of 6.24 years should be offered.

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