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 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.)
(b) If Accrotime does not want to make refunds on more than 16% of the watches it makes, how long should the guarantee period be (to the nearest month)?
From the problem statement, N(32, 7).
(A)
2 years is equivalent to 24 months. We want to know how many watches will break on average.
P(X < 24) = .1265 = 12.65% (ouch)
(B)
This statement asks that we modify the warranty so that less than 16% of watches are refunded. We use our handy-dandy inverse norm function to calculate the z-score associated with this.
z = -.994 = (x - mu) / sigma
With a little algebra, we get
x = (-.994 * sigma) + mu
x = (-.994 * 7) + 32
x = 25.03 months
This answer makes sense because we know that ~12% of watches break within 24 months.
(a) Well, let's start by figuring out how many standard deviations away from the mean is the duration of the guarantee period. Since the average life of the watches is 32 months and the standard deviation is 7 months, the guarantee period of 24 months is 8 months below the mean.
To find out what percentage of the watches will be replaced, we need to calculate the cumulative probability up to this point using the z-score formula. The z-score is calculated as (x - μ) / σ, where x is the value we want to find the probability for (24 months), μ is the mean (32 months), and σ is the standard deviation (7 months).
So, the z-score for the guarantee period is (24 - 32) / 7 = -1.1429.
Looking up this z-score in a standard normal distribution table, we find that the cumulative probability is approximately 0.1271. This means that Accrotime can expect to replace around 12.71% of total production.
(b) Now, let's figure out the guarantee period that ensures refunds are made on no more than 16% of the watches. To do this, we need to find the z-score that corresponds to a cumulative probability of 0.16.
Using the standard normal distribution table, we find that the z-score is approximately -0.9945.
Now, we can use the z-score formula to find the corresponding value for the guarantee period, which is (x - μ) / σ. Rearranging this formula, we have x = μ + (z * σ), where x is the value we want to find (the guarantee period), μ is the mean (32 months), σ is the standard deviation (7 months), and z is the z-score (-0.9945).
Plugging in the values, we get x = 32 + (-0.9945 * 7) = 24.065.
Rounding this to the nearest month, the guarantee period should be 24 months to ensure refunds are made on no more than 16% of the watches.
(a) To find the percentage of total production that the company expects to replace, we need to calculate the probability that a watch's lifetime is less than or equal to 24 months (2 years).
First, we need to standardize the value of 24 months using the z-score formula:
z = (x - μ) / σ
Where x is the value we want to standardize (24), μ is the mean (32), and σ is the standard deviation (7).
z = (24 - 32) / 7
z = -8 / 7
z ≈ -1.14
Next, we need to find the cumulative probability associated with this z-score. This represents the probability that a watch's lifetime is less than or equal to 24 months.
Using a standard normal distribution table or a calculator, we find that the cumulative probability associated with a z-score of -1.14 is approximately 0.1271.
To find the percentage, we multiply this probability by 100:
0.1271 * 100 ≈ 12.71%
Therefore, the company can expect to replace approximately 12.71% of its total production.
(b) To determine how long the guarantee period should be, we need to find the z-score associated with the desired cumulative probability of 16%.
Using a standard normal distribution table or a calculator, we find that the z-score associated with a cumulative probability of 0.16 is approximately -0.9945.
Next, we use the z-score formula to calculate the corresponding value of x (watch lifetime):
z = (x - μ) / σ
Rearranging the formula and substituting the known values, we have:
x = z * σ + μ
x = -0.9945 * 7 + 32
x ≈ 25.06
Therefore, the guarantee period should be approximately 25 months (rounded to the nearest month) to ensure that the company does not need to make refunds on more than 16% of the watches it makes.
To answer these questions, we need to use the normal distribution and the properties of standard deviations. Let's break down each question:
(a) To find the percentage of total production that the company expects to replace, we need to calculate the probability of a watch deteriorating before the 2-year guarantee period. We know that the average lifetime of the watches is 32 months and the standard deviation is 7 months.
To calculate this probability, we need to find the area under the normal distribution curve to the left of 24 months (2 years), which represents the watches that will deteriorate within the guarantee period. We can use a standard normal distribution table or a calculator to find this value.
Using a calculator or a standard normal distribution table, we find that the z-score of 24 months is calculated as:
z = (x - μ) / σ
z = (24 - 32) / 7
z ≈ -1.14
Looking up the z-score in the Z-table, we find that the area to the left of -1.14 is approximately 0.1271.
To get the percentage of total production, we multiply this probability by 100:
0.1271 * 100 ≈ 12.71%
Therefore, the company expects to replace approximately 12.71% of the total production.
(b) To determine the guarantee period given that the company does not want to make refunds on more than 16% of the watches, we need to find the corresponding z-score for the area under the normal distribution curve to the left of 16% (0.16).
Using a standard normal distribution table or a calculator, we look for the z-score that corresponds to an area of 0.16. In this case, we are looking for the z-score that leaves 16% to the right, which is equivalent to an upper tail area of 1 - 0.16 = 0.84.
Looking up the z-score in the Z-table, we find that the z-score closest to this area is approximately 0.996.
To find the corresponding value for the guarantee period, we rearrange the z-score formula:
z = (x - μ) / σ
We want to find x, so we rearrange the formula as follows:
x = z * σ + μ
x = 0.996 * 7 + 32
x ≈ 39.97
Therefore, the guarantee period should be approximately 40 months (rounded to the nearest month) to ensure that the company does not have to make refunds on more than 16% of the watches it produces.