The length of life of a certain type of washer is approximat

The length of life of a certain type of washer is approximately normally distributed with a mean of 6.2 years and a standard deviation of 1.4 years.
I.If this machine is guaranteed for two years,what is the probability that the machine you purchased will require replacement under guarantee
II.What period of time should the manufacturer give as a guarantee if it is willing to replace only 0.5% of the machines?

ANSWER TO QUESTION I:
A washer will require replacement under guarantee if its length of life is shorter than the guarantee period of 2 years.
It is given that
mean = 6.2 years; and
standard deviation = 1.4 years
Hence, the standard unit (z-score) for the guarantee period of 2 years
= (2 - 6.2) / 1.4
= -3
Since the length of life of the washers is approximately normally distributed, the proportion of washers with lifetime between the mean and 3 standard deviations below the mean is 49.865%.　［这个数字需靠查表得出,或用公式计算.］
Hence the proportion of washers with lifetime shorter than 2 years is:
50% - 49.865% = 0.135%
Therefore the probability that the purchased machine will require replacement under guarantee is 0.00135.
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ANSWER TO QUESTION II:
In order to replace only 0.5% of the machines, there can only be 0.5% of them with a lifetime shorter than the guarantee period.
Therefore, there can only be 49.5% (50% - 0.5%) of the machines with lifetime shorter than their mean lifetime but longer than the guaranteed period.
Since the length of life of the washers is approximately normally distributed, the proportion of 49.5% below the mean represents a standard unit of -2.575. ［这个数字需靠查表得出,或用公式计算.］
It is given that
mean = 6.2 years; and
standard deviation = 1.4 years
Therefore the standard unit of -2.575 can be translated into number of years below the mean as follows:
-2.575 x 1.4 years + 6.2 years = 2.595 years,
Therefore, in order to replace only 0.5% of the machines, the guarantee period given by the manufacturer must not be longer than 2.595 years.