Answer:
To determine the time length of the warranty that ensures only 10% of the coffee makers will be replaced before it expires, you need to consider the expected failure rate of the coffee makers.
Let's assume that the coffee makers have a constant failure rate over time, which is typically modeled using the exponential distribution. The exponential distribution is defined by a parameter called the failure rate (λ), which represents the average number of failures per unit of time.
To ensure that only 10% of the coffee makers fail before the warranty expires, you want to find the warranty duration (t) such that the probability of failure before t is 10%. In mathematical terms, you want to find t such that:
P(X < t) = 0.10
Where X is the time until failure, following the exponential distribution. The cumulative distribution function (CDF) of the exponential distribution is given by:
F(t) = 1 - e^(-λt)
Where F(t) is the probability that X is less than or equal to t.
Now, we can set up the equation:
1 - e^(-λt) = 0.10
Solve for t:
e^(-λt) = 0.90
Take the natural logarithm of both sides:
-λt = ln(0.90)
Now, solve for t:
t = ln(0.90) / (-λ)
To find λ, you need data on the failure rate of the coffee makers. If you have data that indicates the average failure rate per unit of time, you can use that value for λ. If you don't have specific data, you may need to make an assumption based on industry standards or similar products.
Once you have the value of λ, you can calculate the warranty duration (t) that ensures only 10% of the coffee makers fail before the warranty expires.
Step-by-step explanation: