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Imagine you want to find out how your car’s service time compares to others. That’s where the Z-score comes in handy. The Z-score indicates how many standard deviations a particular data point is from the mean.

For instance, in the given exercise, the mean service time is 15 minutes, and a standard deviation is given. If your car's service time is significantly different, the Z-score quantifies this difference.

To finding the Z-score, you'll use the formula: \( Z = \frac{X - \mu}{\sigma} \)
where \(X\) is your service time, \(\mu\) is the mean (15 minutes), and \(\sigma\) is the standard deviation (2.4 minutes).

• Values of \( Z \) around 0 indicate times close to the mean.
• Positive values show it took longer than most cars.
• Negative values mean it was faster.

Understanding these scores is essential when calculating probabilities or setting guarantees like in Fast Auto Service.

Let's break down these core concepts: mean and standard deviation. The mean is simply the average of a set of numbers. In our scenario, it represents the average time for servicing a car, set at 15 minutes.

An average gives a central value, but it doesn't tell you how scattered or spread out the service times really are. That’s where standard deviation comes in.

The standard deviation measures this spread. A lower standard deviation indicates the service times are clustered closely around the mean. A higher standard deviation means more variability.

In our example, the deviation is 2.4 minutes, meaning that most service times are within 2.4 minutes of the average. Using these, Fast Auto Service can better manage wait times and set realistic expectations for customers.

Statistical computation involves crunching the numbers to make sense of data patterns. It’s the backbone of our exercise, where it helps find that maximum guaranteed waiting time for Fast Auto Service.

After understanding the problem, we had to use statistical tables or software to find a Z-score corresponding to 95% of the data. In this context, the Z-score was approximately 1.645.

Using this score, a computation translates the Z-score back into minutes so we can determine a specific wait time. The application of the formula \( X = Z \times \sigma + \mu \) allows us to compute the precise waiting time. This complex procedure joins statistical theory with practical application to solve real-world problems.

Probability helps us understand how likely events are within a certain framework. In a normal distribution, probabilities are used to determine how likely any given set of service times are. This garage wants to be sure only 5% of their customers wait longer than a certain time.

For a normal distribution:

• About 68% of data falls within one standard deviation from the mean.
• 95% is within two standard deviations.
• 99.7% is within three standard deviations.

The problem specifies wanting to limit the wait time for at most 5% of the customers, meaning 95% should be serviced faster. Understanding and applying these probabilities ensures the garage sets an accurate guarantee.