Final answer:
The warranty period for an Eco-cook steamer to ensure that not more than 9% are returned should be 42 months, as calculated using the Z-score for the 91st percentile and the given mean and standard deviation of the failure times.
Step-by-step explanation:
The student wants to know the warranty period for an Eco-cook steamer such that no more than 9% of the steamers will be returned due to failure. With a mean time before failure of 37 months and a standard deviation of 4 months, and assuming the failure times are normally distributed, we first need to determine the Z-score that corresponds to the 91st percentile (100% - 9%).
Using a Z-table or calculator, we can find that the Z-score for the 91st percentile is approximately 1.34. We can then use the Z-score formula:
Z = (X - μ) / σ
Where μ is the mean, σ is the standard deviation, and X is the time in months. Rearranging to solve for X, we get:
X = Z × σ + μ
Therefore, X = 1.34 × 4 + 37 = 5.36 + 37 = 42.36 months. Since warranties are typically given in whole months, we should round down to the nearest whole number, so the warranty period should be set to 42 months to ensure no more than 9% of the steamers are returned due to failure.