The average life of a certain type of small motor is 8 years with a standard deviation of 2 years. The manufacturer replaces free all motors that fail while under warranty. If she is willing to provide a 5-year and the distribution of the motor lives is normal, calculate the percentage of motors that will have to be replaced. Use 3 decimal places and enter your answer as a decimal value

And we can find this probability using the normal standard distribution or Excel:

Step-by-step explanation:

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".  

Solution to the problem

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

Where and

We are interested on this probability

Since all the motors with an average life of lesss than 5 years satisfy the warranty period

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 using the normal standard distribution or Excel:

since proposition states that a number is greater than 25, if is true then the number must actually be greater than 25.

this implies that all possible options, except 24, can be accepted