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The Z-score is a statistical measurement that matters a lot when we deal with normal distributions. It tells you how many standard deviations a particular value is from the mean. Think of it as a way to understand where a specific piece of data falls within a standardized group of data. For instance, for the tire manufacturer who wants 98% of tires to last beyond a certain mileage, the Z-score helps identify precisely which mileage value corresponds to the boundary where only 2% of tires last below it. This means 98% of the tires last longer than this mileage.

To find this out, we look up this 2% in the Z-score table, which tells us we need a Z-score of around -2.05. This Z-score value gives us a clear way to translate this threshold into an actual mileage number.

Standard deviation is a key concept that measures how much variation exists in a dataset. Here, we're talking about the lifespan of tires, which has a standard deviation of 1,120 miles. This number gives us insight into how much the mileage of different tires varies from the average of 67,350 miles.

In simpler terms, the standard deviation tells us how spread out the tire lifetimes are. A small standard deviation would mean most tires have a similar lifespan to the average, while a larger one indicates a greater spread or variability in tire life.

• Smaller deviations show less variety among the values.
• Larger deviations indicate more diversity among the values.

Understanding this helps the manufacturer determine the reliability and predictability of how long their tires will last.

The mean is simply the average value of a dataset. It is found by adding all individual values and dividing by the count of those values. Here, the mean we are dealing with is the average tread life of 67,350 miles. This value represents the mid-point of all the tire mileages we expect.

The mean provides the central value around which the normal distribution is centered. Knowing the mean allows us to anchor our Z-score calculations because the Z-score uses the mean as a baseline for determining how unusual or typical a particular data point is. In the tire example, the mean helps determine the average performance of all tires.

Probability calculations in a normal distribution context are about determining the likelihood of a certain event. When using probability in this tire problem, we calculate to find an advertised tire life such that 98% of tires meet or exceed that lifespan.

Here's how we do it:

• First, know the desired probability (here, 98%) that you want your tires to last at least a certain distance.
• Find the Z-score from a standard normal distribution table that matches the leftover probability (here, 2% in the tail).
• Use this Z-score to calculate the specific mileage using the formula: \( X = \mu + Z \cdot \sigma \).

This process allows you to translate percentage sales pitching ("98% of tires last this long!") into a quantifiable number (like 65,054 miles). It involves a few straightforward steps and uses the inherent properties of the normal distribution to make predictions.