Question

Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 30 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 4 months, 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 12% of the watches it makes, how long should the guarantee period be (to the nearest month)?
_________________ months

Answer 1

Answer:

Given that:

Accrotime researchers have shown that the watches have an average life of 30 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 4 months, and the distribution of lifetimes is normal.

Population average life of Accrotine watches = 30 months, and standard deviation = 4 months

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?

With guarantee of 100% replacement after 2 years, that is, 24 months:

To find percent of total production expected to be replaced, we simply need to find the percentage of watches with expected lifetime less than 24 months:

Pr(X < 24) = Pr(z = Z), where

Z = (X - mu)/s.d = (24 - 30)/4 = -1.5

Pr(X < 24) = Pr(z = -1.5)

Using the standard normal distribution table, we know that Pr(z = -1.5) = 0.0668

So, 6.68% of total production is expected to be replaced.

b) If Accrotime does not want to make refunds on more than 12% of the watches it makes, how long should the guarantee period be (to the nearest month)?

If this proportion of total production is required to be kept below 12% that is, for Pr(z = Z) = 0.1200, from the table we can find z = -1.175

So, for Z = -1.175, we need to find value of X. So, proceeding as follows:

-1.175 = (X - 30)/4

X = -1.175*4 + 30 = 25.3 months that is slightly more than 2 years.

Rounding to nearest month, answer is 25 months.