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Mean lifetime represents the average duration that an event or object will exist. In the context of our exercise, it refers to the average number of kilometers that an automobile tire is expected to last. For the tire problem, the mean lifetime is given as 100,000 km. This value is central to our calculations because it sets the baseline for understanding outcomes. When addressing any problems involving probability and normal distribution, knowing the mean gives us a starting point for identifying how a sample deviates from average performance. To calculate practical estimates and probabilities, always align your understanding of the mean lifetime with real-world expectations and how they might differ. In our case, determining how many tires last beyond the mean helps in assessing quality and performance.

Standard deviation is a key statistic in understanding the variability or dispersion of a dataset. It tells us how much variation exists from the average (mean) value. In our exercise, the standard deviation of the tire lifetimes is given as 10,000 km. This means most tire lifetimes fall within 10,000 km above or below the average of 100,000 km. A smaller standard deviation would mean the lifetimes are closely concentrated around the mean, while a larger one indicates more spread. Understanding this helps us evaluate how consistently the tires perform and how much deviation to expect from their average lifetime.

Standard error, often abbreviated as SEM, measures the accuracy with which a sample represents a population. It is calculated by dividing the standard deviation by the square root of the sample size.In this exercise, the standard error is crucial for determining the precision of our mean lifetime estimation with a sample size of 5000. Calculating the SEM involves:

• The standard deviation, \( \sigma = 10,000 \) km
• The sample size, \( n = 5000 \)
• The formula: \( SEM = \frac{\sigma}{\sqrt{n}} \)

After computing, we find the SEM is approximately 141.42 km. This value gives us insight into how much the sample mean might differ from the true population mean, allowing for more precise statistical testing.

The Z-score is a statistical measure that describes a value's position relative to the mean of a group of values. It shows how many standard deviations an element is from the mean.To find the Z-score in this context, we use the formula:\[Z = \frac{\bar{x} - \mu}{SEM}\]Here, \( \bar{x} \) is our sample mean (100,282 km), \( \mu \) is the population mean (100,000 km), and SEM is the standard error (141.42 km). After substituting the given values, the Z-score is approximately 1.993. Looking up this Z-score in a standard normal distribution table yields a cumulative probability of 0.9767. This indicates that 97.67% of the tire samples have a mean lifetime less than 100,282 km, providing a clear idea of how rare it is to surpass that mean. Subtracting this percentage from 100% gives the proportion of samples exceeding this mean conventional for decision-making processes and quality controls.