Question

Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 32 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.

(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

We can let the life of the watch be represented by 'X'.

We then use normal probabilties table and the normal percentage table.

(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.)

Percentage here means we need to calculate the probabilities.

If the watch lasts for more than 2 years then no refund but if less than 2 years then refund has to be paid. Therefore the % to replace is given by

P(X < 2 years) = P(X < 24) ...................... working with months

= P(Z < -1.6)

= 1 - P(Z < 1.6)

= 1 - 0.9452

= 0.0548

The company can expect watches 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 they want to expect only 12% of replacements then 'x' months should be guaranteed above which no replacements would be made.

P(X < x) = 12%

P(Z < ) = 12%

P(Z > -_) = 12%

We changed the sign because the normal percentage tables which we are going to use provides for greater than probbilties and for p < 50%.

= 1.175

x = 26.13