1. suppose the failure time of a certain mechanical component is normally distributed with mean value of 10 years and standard deviation of 2 years. suppose a mechanical component of above type is randomly selected. what is the probability that the failure time of the component is (a) between 8 years to 12 years. (b) more than 13 years.

a)

b)

Step-by-step explanation:

1) Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

2) Part a

Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability on this way:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

3) Part b

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can use the complement rule to find this probability:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.