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Z-scores, often known as standard scores, are a way to describe a data point's distance from the mean of a distribution. The z-score is expressed in terms of standard deviations and is a handy tool for understanding where a particular value lies within a normal distribution. Whenever you calculate a z-score, you're essentially asking, "How many standard deviations away is this value from the mean?"

In a normal distribution:

• A z-score of 0 means the data point is exactly at the mean.
• A positive z-score indicates that the data point is higher than the mean.
• A negative z-score shows that the data point is below the mean.

These scores are crucial when you need to solve for unknowns like the mean (\(\mu\)) or the standard deviation (\(\sigma\)), as seen in this exercise. By using z-scores, we solve the percentiles given by the problem (like 5% and 10%) to set up equations that allow us to derive these unknown parameters.

The mean and standard deviation are two fundamental concepts in statistics, especially when dealing with normal distributions. The mean (\(\mu\)) is the average value of all data points in a distribution and serves as a central point around which data is dispersed. Calculating the mean involves summing up all speeds and dividing by the number of data points.

The standard deviation (\(\sigma\)) tells us how much individual data points vary or deviate from the mean. A smaller standard deviation indicates data points are close to the mean, while a larger one suggests a wider spread. In normal distributions, like our vehicle speeds, approximately 68% of the data falls within one standard deviation from the mean, 95% within two, and about 99.7% within three.

In the context of the problem, understanding these concepts helps us calculate the parameters using given percentile values and solve equations with z-scores. It allows us to model the vehicle speeds accurately based on the distribution's characteristics.

Probability calculations in a normal distribution allow us to find the likelihood of a random variable falling within a certain range. This is particularly useful when you have a bell-shaped curve where most outcomes are clustered around a mean, such as vehicle speeds on a bridge.

To determine probabilities:

• First, identify the relevant z-scores for given values.
• Use standard normal distribution tables or calculators to find probabilities associated with these z-scores.
• Combine these probabilities to find the overall probability within or outside a range.

For example, if we want to know the probability that a vehicle's speed is between 50 and 65 m/h, we calculate the z-scores for these speeds and look up the probability of occurrences between these values. Also, to find the likelihood of speeds exceeding a specific limit, like 70 m/h, we need a similar approach by finding the complement of the cumulative probability at that z-score.

This process helps us understand the distribution's behavior, providing insights into vehicle flow and speed regulation on the bridge.