Question

1. Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have determined that the watches have an average life of 55 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 watches for 4 years (48 months) after purchase, what percentage of total production will the company expect to replace?

b. If Accrotime only wants to refund 4% (or less) of the watches it makes, how long should the guarantee period be (to the nearest month).

Answer 1

Let X denote the lifetime of the watches and according to the question, X~N(55, 5²=25).

a) Now, if Accrotime guarantees a full refund on watches for 4 years(48 months), the proportion of watches that the company will expect to replace is,

P(X48)

= P[(X-55)/5(48-55) /5]

=P(Z-1.4) [where, Z=(X-55) /5~N(0, 1) ]

=0.0808

Thus, the company will expect 0.0808*100=8.08% of total production to replace.

b) let the guarantee period of a watch be x months so that Accrotime expects to refund 4%(or less) of the watches it makes.

Thus, P(X<x) =0.04(taking the equality)

Or, P[(X-55) /5<(x-55) /5]=0.04

Or, P[Z<(x-55)/5]=0.04

From, standard normal table, (x-55)/5=-1.76, or x=46.2.

Thus, the guarantee period should be 46.2 months.