Paper For Above instruction
The problem at hand involves determining a specific mileage threshold below which only the shortest 1% of tire lifetimes lie. These lifetimes are modeled as a normally distributed random variable, X, with known mean and standard deviation: µ = 40,000 miles and σ = 2,000 miles. The goal is to find the cutoff value, x, such that only 1% of all tires fall below this mileage, corresponding to the first percentile of the distribution. This is a typical application of the properties of the normal distribution and its related z-scores.
To approach this, we first recognize that in a normal distribution, the percentile point associated with the lowest 1% can be found using the standard normal distribution table or z-score calculations. The critical step is identifying the z-score corresponding to the 1% lower tail of the standard normal distribution. Based on standard normal tables, the z-score associated with the 1% quantile (or the 0.01 cumulative probability) is approximately -2.33.
Using the z-score formula:
z = (x - µ) / σ
we can rearrange this to solve for x:
x = µ + z * σ
Plugging in the known values:
x = 40,000 + (-2.33) * 2,000
we obtain:
x = 40,000 - 4,660 = 35,340
Therefore, the largest value for a tire's lifetime that still falls within the shortest 1% is approximately 35,340 miles.
This means the manufacturer should set the mileage guarantee at about 35,340 miles so that only the bottom 1% of tires, those that last less than this distance, are covered under free replacement. This approach balances customer satisfaction with cost management, as only the tires with the shortest expected life are replaced without charge. The application of the normal distribution is crucial here, enabling the manufacturer to statistically determine a cutoff point that aligns with their policy goals.
The overall logic hinges on the properties of the normal distribution—the symmetry and known percentiles—and illustrates the practical use of z-scores in real-world quality assurance and warranty policies. It is also a testament to the utility of statistical methods in establishing fair and economical business practices based on probabilistic understanding of product lifetimes.
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