Suppose that a company guarantees refrigerators and will replace with a new one that breaks while it is under the guarantee. However, the company does not want to replace more than 5% of the refrigerators under the guarantee. For how long should the guarantee be made? You can round to the nearest tenth of a year.
Don't you think you need some data on the expected lifetime of the units?
Consumer Reports indicated that the average life of a refrigerator is 13 years with a standard deviation of 2.5 years. (Assume the distribution of the age of the refrigerators is approximately normal.)
To solve this problem, we need to find the length of time for the guarantee such that the number of refrigerators expected to break is within 5% of the total number of refrigerators.
Let's break down the problem step by step:
Step 1: Determine the percentage of refrigerators expected to break within the guarantee period.
Since the company does not want to replace more than 5% of the refrigerators, we need to find the percentage of defective refrigerators that will be covered by the guarantee. Let's assume this percentage is "P."
Step 2: Calculate the expected number of refrigerators that will break.
The next step is to determine the expected number of refrigerators that will break. We can do this by multiplying the percentage of defective refrigerators (P) by the total number of refrigerators. Let's assume the total number of refrigerators is "N."
Expected number of refrigerators that will break = P * N
Step 3: Set the expected number of broken refrigerators equal to 5% of the total number of refrigerators.
Since the company does not want to replace more than 5% of the refrigerators, we need to set the expected number of broken refrigerators equal to 5% of the total number of refrigerators.
P * N = 0.05 * N
Step 4: Solve for P.
To find the value of P, we divide both sides of the equation by N.
P = 0.05
Step 5: Calculate the length of the guarantee period.
Now that we know P is equal to 0.05, the percentage of defective refrigerators, we can find the duration of the guarantee by dividing 1 by P (1/0.05).
Length of the guarantee period = 1 / P = 1 / 0.05 = 20
Therefore, the guarantee should be made for 20 years.
Note: This calculation assumes a constant percentage of defective refrigerators over time and does not take into account potential changes in the company's business model or industry standards.