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The standard normal distribution is a key concept in statistics that helps handle data that follows a pattern. It takes any normal distribution and converts it into a special normal distribution with a mean of 0 and a standard deviation of 1.
This is mainly done using the Z-score. Understanding the characteristics of the standard normal distribution can make statistical calculations a lot easier. The bell curve shape of the distribution helps visualize how data clusters around the mean. With most data falling within one standard deviation from the mean, it reflects how respective probabilities are spread over the values.
By converting any normal distribution to a standard normal, you can use a Z-table, simplifying the process of looking up probabilities without needing to recalculate them every time. In our scenario, the time it takes to find a parking space at 9 A.M. is normally distributed with given mean and standard deviation. By converting this distribution to a standard normal one, we can easily calculate the required probabilities.

Calculating probabilities in a normal distribution requires an understanding of how likely certain outcomes are. With a normal distribution, these probabilities often concern questions like the likelihood of a random variable being less than or greater than a particular value. Probabilities in a normal distribution are determined with the help of the Z-table. Once the Z-score for a particular point in the distribution is calculated, you can reference the Z-table to find the probability that a value is below the Z-score. This allows you to determine how big or small the likelihood is for a specific event.
In probability terms, the area under the curve of the distribution graph tells us about these probabilities. A larger area signifies a higher probability and vice versa. In the exercise, we're finding the probability of the time taken to find a parking spot being less than one minute. The calculated probability from the Z-table turned out to be 0.0228, indicating a rare occurrence. This low probability confirms our surprise if such an event occurs.

Z-Score conversion is a critical process in standardizing normal distributions. This conversion helps translate the original data point in a distribution to one that fits the standard normal distribution. By doing so, it quantifies how far and in what direction a data point deviates from the mean.The formula to compute the Z-score is: \[ Z = \frac{X - \mu}{\sigma} \]Here, X represents the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. The Z-score tells you how many standard deviations an element is from the mean. Positive Z-scores indicate values above the mean, while negative values suggest those below.In our solution, we calculated the Z-score for finding a parking spot in less than 1 minute, resulting in a Z-score of -2. This score translates to the point being 2 standard deviations below the mean. Once calculated, it became straightforward to use the Z-table to find the associated probability.