Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 26 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 7 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.)
To find the percentage of total production that the company is expected to replace, we need to determine the area under the normal distribution curve for the watches that have a lifetime less than or equal to 24 months, which is equivalent to 2 years.
First, we need to convert the given information into a standard normal distribution by using the z-score formula:
Z = (X - μ) / σ
Where:
Z is the z-score
X is the value we are interested in (in this case, 24 months)
μ is the mean (average) watch lifetime (26 months)
σ is the standard deviation (7 months)
Z = (24 - 26) / 7 = -2 / 7 ≈ -0.286
Now, we'll use a standard normal distribution table or a calculator to find the area under the curve to the left of the z-score of -0.286. This represents the percentage of watches that have a lifetime less than or equal to 24 months.
Looking up the z-score of -0.286 in the Z-table, we find that the corresponding area is approximately 0.3889.
However, since we want the area to the left of the z-score, we need to calculate 0.5 - 0.3889 = 0.1111.
So, approximately 11.11% of the watches will have a lifetime less than or equal to 24 months, and the company is expected to replace this percentage of the total production.