Question

Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 36 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 5months, and the distribution of lifetimes is normal.

(a) If Accrotime guarantees a full refund on any defective watch for 2 years after purchase, what percentage of total production will the company expect to replace? (Round your answer to two decimal places.)
______%

(b) If Accrotime does not want to make refunds on more than 6% of the watches it makes, how long should the guarantee period be (to the nearest month)?
____ months

Answer 1

Solution:

We are given that: Accrotime researchers have shown that the watches have an average life of 36 months before certain electronic components deteriorate, causing the watch to become unreliable.

The standard deviation of watch lifetimes is 5 months, and the distribution of lifetimes is normal.

Thus Mean = and

Part a) If Accrotime guarantees a full refund on any defective watch for 2 years after purchase, what percentage of total production will the company expect to replace?

For 2 years , number of months = 2 * 12 = 24 months.

that is we have to find:

P( X < 24) = .................?

Thus find z score for x = 24

Thus we get:

P( X < 24) =P( Z < -2.40)

Look in z table for z = -2.4 and 0.00 and find corresponding area

thus we get:

P( Z < -2.40) = 0.0082

Thus

P( X < 24) =P( Z < -2.40)

P( X < 24) = 0.0082

P( X < 24) = 0.82%

Part b) If Accrotime does not want to make refunds on more than 6% of the watches it makes, how long should the guarantee period be (to the nearest month)?

That is we have to find x value such that:

P( X < x ) =6%

P( X < x ) =0.06

Thus find z value such that:

P( Z < z ) =0.06

Look in z table for area = 0.0600 or its closest area and find corresponding z value.

Area 0.0600 is between 0.0594 and 0.0606 and both the area are at same distance from 0.0600

thus we find both z values and then find their average.

For Area 0.0594 , z value is -1.56
For Area 0.0606 , z value is -1.55

Thus average of z values is :

z = ( (-1.55) + (-1.56) ) / 2 = ( -3.11 )/ 2 = -1.555

Thus required z value is : z = -1.555

Now use following formula to find x value:

months.