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 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
mean is 32 , s.d. is 4 ... use a z-score table to find the answers
(a) 2 yr is 2 s.d. below the mean ... .023%
(b) 12% of the population is approx. 2.257 s.d. below the mean
... (32 months) - [(4 months) * 2.257) = ?
To answer these questions, we need to use the concept of the normal distribution and z-scores. Here's how we can find the answers step by step:
(a) To find the percentage of total production that Accrotime expects to replace, we need to determine the proportion of watches that have a lifetime less than or equal to 24 months (2 years).
Step 1: Calculate the z-score for 24 months using the formula: z = (x - μ) / σ, where x is the value of interest, μ is the mean, and σ is the standard deviation. In this case, x = 24 months, μ = 32 months, and σ = 4 months.
z = (24 - 32) / 4 = -2
Step 2: Use a z-table or a calculator with a normal distribution function to find the proportion associated with the z-score of -2. The z-table gives us the proportion of values below a certain z-score.
Looking up the z-score of -2 in the z-table, we find that the proportion associated with it is approximately 0.0228.
Step 3: Convert the proportion to a percentage by multiplying it by 100.
Percentage = 0.0228 * 100 = 2.28%
Therefore, Accrotime can expect to replace approximately 2.28% of the total production.
(b) To find the guarantee period that ensures Accrotime doesn't have to make refunds on more than 12% of the watches, we need to determine the lifetime value corresponding to the z-score associated with the 12th percentile. In other words, we want to find the x-value that corresponds to a proportion of 0.12 below it.
Step 1: Lookup the z-score from the z-table that corresponds to a proportion of 0.12 below it. In this case, we are looking for the z-score with an area of 0.12 to the left.
The z-score we find is approximately -1.18.
Step 2: Use the reverse of the z-score formula to solve for x: x = μ + (z * σ). Here, μ = 32 months, σ = 4 months, and z = -1.18.
x = 32 + (-1.18 * 4) = 32 - 4.72 = 27.28
Therefore, the guarantee period should be approximately 27 months (rounded to the nearest month) to ensure that Accrotime does not have to make refunds on more than 12% of the watches it makes.