Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 36 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.)
4.33
Correct: Your answer is correct.
%
(b) If Accrotime does not want to make refunds on more than 9% of the watches it makes, how long should the guarantee period be (to the nearest month)?
Incorrect: Your answer is incorrect.
months
To solve part (a) of the problem, we need to find the percentage of total production that the company expects to replace. We are given that the average lifetime of the watches is 36 months and the standard deviation is 7 months. Since the distribution of lifetimes is normal, we can use the normal distribution to calculate probabilities.
To find the percentage of watches that will last less than 2 years (24 months), we need to calculate the probability that a randomly selected watch has a lifetime less than 24 months.
First, we need to standardize the value 24 by subtracting the mean (36) and dividing by the standard deviation (7):
z = (24 - 36) / 7
z = -1.71
Next, we use a standard normal distribution table (or a calculator or software) to find the cumulative probability to the left of -1.71.
From the z-table, we find that the probability (P) for z = -1.71 is approximately 0.0423.
Since we want the percentage, we multiply this probability by 100 to get:
Percentage = 0.0423 * 100 = 4.23%
Therefore, the company expects to replace approximately 4.23% of the total production.
Now, let's move on to part (b).
In this part, we want to determine the guarantee period that ensures the company does not have to make refunds on more than 9% of the watches. We need to find the value of the lifetime (in months) that corresponds to the cumulative probability of 9% to the left.
Using a standard normal distribution table (or a calculator or software), we find that the closest cumulative probability to 9% is a z-value of approximately -1.34.
We can then find the corresponding lifetime (in months) using the standardization formula:
z = (x - mean) / standard deviation
Rearranging the formula to solve for x, we have:
x = z * standard deviation + mean
x = -1.34 * 7 + 36
x ≈ 25.42
Thus, to ensure that the company does not have to make refunds on more than 9% of the watches, the guarantee period should be approximately 25 months (rounded to the nearest month).